Neural-PDE-Solver

PDE: Partial Differentiable Equation

Contributed by Chunyang Zhang.

Content

1. Survey
2. Models
2.1 PINN	2.2 DeepONet
2.3 Fourier Operator	2.4 Graph Network
2.5 Green Function	2.6 Finite Element
2.7 Convolution	2.8 AutoEncoder
2.9 Neural Operator	2.10 Identification
2.11 Machine Learning	2.12 Neural ODE
3. Mechanism
3.1 Library	3.2 Analysis
3.3 Attention	3.4 Neural Implicit Flow
3.5 Disentangle	3.6 Meta Learning
3.7 AutoML	3.8 Sampling
3.9 Latent Space	3.10 Loss Function
3.11 Decomposition	3.12 Mesh
3.13 Generative Model	3.14 Gaussian Process
3.15 Solver	3.16 Variation
3.17 Bayesian	3.18 Lagrangian
3.19 Uncertainty Quantification	3.20 Active Learning
3.21 Active Learning	3.22 Multi Scale
3.23 Multi Fidelity	3.24 Multi Grid
4. Applications
4.1 Optimization	4.2 Fluid
4.3 Reconstruction	4.4 Physics
4.5 Image	4.6 Mechanics
4.7 Robotics	4.8 Cybernetics
4.9 Inverse Design	4.10 Quantum
4.11 Climate	4.12 Game Theory
4.13 Manufacturing

Survey Papers

Physics-informed machine learning. Nature Reviews Physics, 2021. paper

George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang.
Neural operator: Learning maps between function spaces. arXiv, 2021. paper

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.
Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Physical Review Fluids, 2018. paper

Jinlong Wu, Heng Xiao, and Eric Paterson.
Integrating scientific knowledge with machine learning for engineering and environmental systems. ACM Computing Surveys, 2023. paper

Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar.
Physical laws meet machine intelligence: Current developments and future directions. Artificial Intelligence Review, 2022. paper

Temoor Muther, Amirmasoud Kalantari Dahaghi, Fahad Iqbal Syed, and Vuong Van Pham.
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswami, Zhongqiang Zhang, and George Em Karniadakis.
Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Beyond Traditional AI: The Impact of Machine Learning on Scientific Computing, 2022. book

MingyuanYang and John T.Foster
When physics meets machine learning: A survey of physics-informed machine learning. arXiv, 2022. paper

Chuizheng Meng, Sungyong Seo, Defu Cao, Sam Griesemer, and Yan Liu.
Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing. arXiv, 2022. paper

Salah A. Faroughi, Nikhil M. Pawar, C´elio Fernandes, Subasish Das, Nima K. Kalantari, and Seyed Kourosh Mahjour.
Physics-informed machine learning: A survey on problems, methods and applications. arXiv, 2022. paper

Zhongkai Hao, Songming Liu, Yichi Zhang, Chengyang Ying, Yao Feng, Hang Su, and Jun Zhu.
An overview on deep learning-based approximation methods for partial differential equations. arXiv, 2020. paper

Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, and Benno Kuckuck.
Three ways to solve partial differential equations with neural networks—A review. GAMM‐Mitteilungen, 2021. paper

Jan Blechschmidt and Oliver G. Ernst.
Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review. GAMM‐Mitteilungen, 2021. paper

Alexander Heinlein, Axel Klawonn, Martin Lanser, and Janine Weber.
Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing. arXiv, 2022. paper

Salah A Faroughi, Nikhil Pawar, Celio Fernandes, Subasish Das, Nima K. Kalantari, and Seyed Kourosh Mahjour.
Partial differential equations meet deep neural networks: A survey. arXiv, 2022. paper

Shudong Huang, Wentao Feng, Chenwei Tang, and Jiancheng Lv.

Models

PINN

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 2020. paper

Raissi Maziar, Alireza Yazdani, and George Em Karniadakis.
Deep hidden physics models: Deep learning of nonlinear partial differential equations. JMLR, 2018. paper

Maziar Raissi.
A universal PINNs method for solving partial differential equations with a point source. IJCAI, 2022. paper

Xiang Huang, Hongsheng Liu, Beiji Shi, Zidong Wang, Kang Yang, Yang Li, Min Wang, Haotian Chu, Jing Zhou, Fan Yu, Bei Hua, Bin Dong, and Lei Chen.
Parallel physics-informed neural networks via domain decomposition. JCP, 2021. paper

Khemraj Shukla, Ameya D.Jagtap, and George Em Karniadakis.
Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs. Computer Methods in Applied Mechanics and Engineering, 2023. paper

Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh.
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering, 2022. paper

N.Sukumar and Ankit Srivastava.
Physics-informed multi-LSTM networks for meta-modeling of nonlinear structures. Computer Methods in Applied Mechanics and Engineering, 2020. paper

Ruiyang Zhang, Yang Liu, and Hao Sun.
Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis.
Multi-output physics-informed neural networks for forward and inverse PDE problems with uncertainties. Computer Methods in Applied Mechanics and Engineering, 2022. paper

MingyuanYang and John T.Foster
PPINN: Parareal physics-informed neural network for time-dependent PDEs. Computer Methods in Applied Mechanics and Engineering, 2020. paper

Xuhui Meng, Zhen Li, Dongkun Zhang, and George Em Karniadakis.
CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Pao-Hsiung Chiu, Jian Cheng Wong, Chinchun Ooi, My Ha Dao, and Yew-Soon Ong.
Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Thomas O’Leary-Roseberry, Umberto Villa, Peng Chen, and Omar Ghattas.
Physics-augmented learning: A new paradigm beyond physics-informed learning. NIPS, 2021. paper

Ziming Liu, Yuanqi Du, Yunyue Chen, and Max Tegmark.
Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm. Physics Letters A, 2021. paper

Yifan Mo, Liming Ling, and Delu Zeng.
Solving Benjamin–Ono equation via gradient balanced PINNs approach. The European Physical Journal Plus, 2022. paper

Xiangyu Yang and Zhen Wang.
Robust learning of physics informed neural networks. arXiv, 2021. paper

Chandrajit Bajaj, Luke McLennan, Timothy Andeen, and Avik Roy.
Learning physics-informed neural networks without stacked back-propagation. arXiv, 2022. paper

Di He, Wenlei Shi, Shanda Li, Xiaotian Gao, Jia Zhang, Jiang Bian, Liwei Wang, and Tieyan Liu.
NeuralPDE: Automating physics-informed neural networks (PINNs) with error approximations. arXiv, 2021. paper

Kirill Zubov, Zoe McCarthy, Yingbo Ma, Francesco Calisto, Valerio Pagliarino, Simone Azeglio, Luca Bottero, Emmanuel Luján, Valentin Sulzer, Ashutosh Bharambe, Nand Vinchhi, Kaushik Balakrishnan, Devesh Upadhyay, and Chris Rackauckas.
Physics informed RNN-DCT networks for time-dependent partial differential equations. ICCS, 2022. paper

Benjamin Wu, Oliver Hennigh, Jan Kautz, Sanjay Choudhry, and Wonmin Byeon.
Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation. JCP, 2022. paper

Amirhossein Arzani, Kevin W.Cassel, and Roshan M.D'Souza.
A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. JCP, 2022. paper

Lei Yuan, Yiqing Ni, Xiangyun Deng, and Shuo Hao.
A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Shahed Rezaei, Ali Harandi, Ahmad Moeineddin, Baixiang Xua, and Stefanie Reese.
Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations. Computers & Mathematics with Applications, 2023. paper

Siping Tang, Xinlong Feng, Wei Wu, and Hui Xu.
A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Revanth Mattey and Susanta Ghosh.
RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations. Computers and Fluids, 2022. paper

Pai Peng, Jiangong Pan, Hui Xu, and Xinlong Feng.
A-WPINN algorithm for the data-driven vector-soliton solutions and parameter discovery of general coupled nonlinear equations. Physica D: Nonlinear Phenomena, 2022. paper

Shumei Qin, Min Li, Tao Xu, and Shaoqun Dong.
Physics-informed neural networks with adaptive localized artificial viscosity. arXiv, 2022. paper

E.J.R. Coutinho, M. Dall'Aqua, L. McClenny, M. Zhong, U. Braga-Neto, and E. Gildin.
Physics-informed neural operator for learning partial differential equations. arXiv, 2021. paper

Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar.
Anisotropic, sparse and interpretable physics-informed neural networks for PDEs. arXiv, 2022. paper

Amuthan A. Ramabathiran and Prabhu Ramachandran.
Fast neural network based solving of partial differential equations. arXiv, 2022. paper

Jaroslaw Rzepecki, Daniel Bates, and Chris Doran.
Discontinuity computing using physics-informed neural network. arXiv, 2022. paper

Li Liu, Shengping Liu, Hui Xie, Fansheng Xiong, Tengchao Yu, Mengjuan Xiao, Lufeng Liu, and Heng Yong.
Learning differentiable solvers for systems with hard constraints. arXiv, 2022. paper

Geoffrey Négiar, Michael W. Mahoney, and Aditi S. Krishnapriyan.
Momentum diminishes the effect of spectral bias in physics-informed neural networks. arXiv, 2022. paper

Ghazal Farhani, Alexander Kazachek, and Boyu Wang.
Δ-PINNs: Physics-informed neural networks on complex geometries. arXiv, 2022. paper

Francisco Sahli Costabal, Simone Pezzuto, and Paris Perdikaris.
Replacing automatic differentiation by Sobolev Cubatures fastens physics informed neural nets and strengthens their approximation power. arXiv, 2022. paper

Juan Esteban Suarez Cardona and Michael Hecht.
FO-PINNs: A first-order formulation for physics informed neural networks. arXiv, 2022. paper

Rini J. Gladstone, Mohammad A. Nabian, and Hadi Meidani.
Augmented physics-informed neural networks (APINNs): A gating network-based soft domain decomposition methodology. arXiv, 2022. paper

Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, and Kenji Kawaguchi.
Physics-informed neural networks for operator equations with stochastic data. arXiv, 2022. paper

Paul Escapil-Inchauspé and Gonzalo A. Ruz.
Physics-informed neural networks with unknown measurement noise. arXiv, 2022. paper

Philipp Pilar and Niklas Wahlstrom.
On the compatibility between a neural network and a partial differential equation for physics-informed learning. arXiv, 2022. paper

Kuangdai Leng and Jeyan Thiyagalingam.
Pre-training strategy for solving evolution equations based on physics-informed neural networks. arXiv, 2022. paper

Jiawei Guo, Yanzhong Yao, Han Wang, and Tongxiang Gu.
L-HYDRA: Multi-head physics-informed neural networks. arXiv, 2023. paper

Zongren Zou and George Em Karniadakis.
PINN for dynamical partial differential equations is not training deeper networks rather learning advection and time variance. arXiv, 2023. paper

Siddharth Rout.
Wavelets based physics informed neural networks to solve non-linear differential equations. Scientific Reports, 2023. paper

Ziya Uddin, Sai Ganga, Rishi Asthana, and Wubshet Ibrahim.
Improved training of physics-informed neural networks using energy-based priors: A study on electrical impedance tomography. ICLR, 2023. paper

Akarsh Pokkunuru, Pedram Rooshenas, Thilo Strauss, Anuj Abhishek, and Taufiquar Khan.
Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems. arXiv, 2023. paper

Sarah Perez, Suryanarayana Maddu, Ivo F. Sbalzarini, and Philippe Poncet.
Efficient physics-informed neural networks using hash encoding. arXiv, 2023. paper

Xinquan Huang and Tariq Alkhalifah.
Ensemble learning for physics informed neural networks: A gradient boosting approach. arXiv, 2023. paper

Zhiwei Fang, Sifan Wang, and Paris Perdikaris.
On the limitations of physics-informed deep learning: Illustrations using first order hyperbolic conservation law-based traffic flow models. arXiv, 2023. paper

Archie J. Huang and Shaurya Agarwal.
Achieving high accuracy with PINNs via energy natural gradients. arXiv, 2023. paper

Johannes Müller and Marius Zeinhofer.
Implicit stochastic gradient descent for training physics-informed neural networks. arXiv, 2023. paper

Ye Li, Songcan Chen, and Shengjun Huang.
NSGA-PINN: A multi-objective optimization method for physics-informed neural network training. arXiv, 2023. paper

Binghang Lu, Christian B. Moya, and Guang Lin.
Improving physics-informed neural networks with meta-learned optimization. arXiv, 2023. paper

Alex Bihlo.
MetaPhysiCa: OOD robustness in physics-informed machine learning. arXiv, 2023. paper

S Chandra Mouli, Muhammad Ashraful Alam, and Bruno Ribeiro.
HomPINNs: Homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions. arXiv, 2023. paper

Haoyang Zheng, Yao Huang, Ziyang Huang, Wenrui Hao, and Guang Lin.
iPINNs: Incremental learning for physics-informed neural networks. arXiv, 2023. paper

Aleksandr Dekhovich, Marcel H.F. Sluiter, David M.J. Tax, and Miguel A. Bessa.
Global convergence of deep Galerkin and PINNs methods for solving partial differential equations. arXiv, 2023. paper

Francisco Eiras, Adel Bibi, Rudy Bunel, Krishnamurthy Dj Dvijotham, Philip Torr, and M. Pawan Kumar.
Provably correct physics-informed neural networks. arXiv, 2023. paper

Deqing Jiang, Justin Sirignano, and Samuel N. Cohen.

DeepONet

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. NMI, 2021. paper

Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis.
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. SA, 2021. paper

Wang Sifan, Hanwen Wang, and Paris Perdikaris.
Deep transfer operator learning for partial differential equations under conditional shift. NMI, 2022. paper

Somdatta Goswami, Katiana Kontolati, Michael D. Shields, and George Em Karniadakis.
Variable-input deep operator networks. arXiv, 2022. paper

Michael Prasthofer, Tim De Ryck, and Siddhartha Mishra.
MIONet: Learning multiple-input operators via tensor product. arXiv, 2022. paper

Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis.
Error estimates for DeepONets: A deep learning framework in infinite dimensions. Transactions of Mathematics and Its Applications, 2022. paper

Samuel Lanthaler, Siddhartha Mishra, and George E Karniadakis.
Long-time integration of parametric evolution equations with physics-informed DeepONets. arXiv, 2021. paper

Sifan Wang and Paris Perdikaris.
Improved architectures and training algorithms for deep operator networks. Journal of Scientific Computing, 2022. paper

Sifan Wang, Hanwen Wang, and Paris Perdikaris.
SVD perspectives for augmenting DeepONet flexibility and interpretability. arXiv, 2022. paper

Simone Venturi and Tiernan Casey.
Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs. arXiv, 2021. paper

Guang Lin, Christian Moya, and Zecheng Zhang.
Bi-fidelity modeling of uncertain and partially unknown systems using DeepONet. arXiv, 2022. paper

Subhayan De, Matthew Reynolds, Malik Hassanaly, Ryan N. King, and Alireza Doostan.
MultiAuto-DeepONet: A multi-resolution autoencoder DeepONet for nonlinear dimension reduction, uncertainty quantification and operator learning of forward and inverse stochastic problems. arXiv, 2022. paper

Jiahao Zhang, Shiqi Zhang, and Guang Lin.
Transfer learning enhanced DeepONet for long-time prediction of evolution equations. arXiv, 2022. paper

Wuzhe Xu, Yulong Lu, and Li Wang.
B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD. JCP, 2023. paper

Guang Lin, Christian Moy, and Zecheng Zhang.
VB-DeepONet: A Bayesian operator learning framework for uncertainty quantification. Engineering Applications of Artificial Intelligence, 2023. paper

Shailesh Garg and Souvik Chakraborty.

Fourier Operator

Fourier neural operator for parametric partial differential equations. ICLR, 2021. paper

Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.
On universal approximation and error bounds for Fourier neural operators. JMLR, 2021. paper

Nikola Kovachki, Samuel Lanthaler, and Siddhartha Mishra.
HyperFNO: Improving the generalization behavior of Fourier neural operators. NIPS, 2022. paper

Francesco Alesiani, Makoto Takamoto, and Mathias Niepert.
Neural operator: Graph kernel network for partial Differential equations. arXiv, 2020. paper

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.
Fourier neural operator with learned deformations for PDEs on general geometries. arXiv, 2022. paper

Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar.
Multipole graph neural operator for parametric partial differential equations. NIPS, 2020. paper

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.
Fast sampling of diffusion models via operator learning. NIPS, 2022. paper

Hongkai Zheng, Weili Nie, Arash Vahdat, Kamyar Azizzadenesheli, and Anima Anandkumar.
Factorized Fourier neural operators. ICLR, 2023. paper

Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong.
Model inversion for spatio-temporal processes using the Fourier neural operator. NIPS, 2023. paper

Dan MacKinlay, Dan Pagendam, Petra M. Kuhnert, Tao Cui, David Robertson, and Sreekanth Janardhanan.
Learning deep implicit Fourier neural operators (IFNOs) with applications to heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Huaiqian You, Quinn Zhang, Colton J. Ross, Chung-Hao Lee, and Yue Yu.
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Sifan Wang, Hanwen Wang, and Paris Perdikaris.
Semi-supervised learning of partial differential operators and dynamical flows. arXiv, 2022. paper

Michael Rotman, Amit Dekel, Ran Ilan Ber, Lior Wolf, and Yaron Oz.
Non-equispaced Fourier neural solvers for PDEs. arXiv, 2022. paper

Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z, and Li Cari.
Incremental spectral learning Fourier neural operator. arXiv, 2022. paper

Jiawei Zhao, Robert Joseph George, Yifei Zhang, Zongyi Li, and Anima Anandkumar.
Fourier continuation for exact derivative computation in physics-informed neural operators. arXiv, 2022. paper

Haydn Maust, Zongyi Li, Yixuan Wang, Daniel Leibovici, Oscar Bruno, Thomas Hou, and Anima Anandkumar.
Non-equispaced Fourier neural solvers for PDEs. arXiv, 2023. paper

Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z, and Li Cari.
Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors. arXiv, 2023. paper

Bilal Thonnam Thodi, Sai Venkata Ramana Ambadipudi, and Saif Eddin Jabari.
Domain agnostic Fourier neural operators. arXiv, 2023. paper

Ning Liu, Siavash Jafarzadeh, and Yue Yu.

Graph Network

Message passing neural PDE solvers. ICLR, 2022. paper

Johannes Brandstetter, Daniel E. Worrall, and Max Welling.
Predicting physics in mesh-reduced space with temporal attention. ICLR, 2022. paper

Xu Han, Han Gao, Tobias Pfaff, Jianxun Wang, and Liping Liu.
Learning mesh-based simulation with graph networks. ICLR, 2021. paper

Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter Battaglia.
Learning large-scale subsurface simulations with a hybrid graph network simulator. KDD, 2022. paper

Tailin Wu, Qinchen Wang, Yinan Zhang, Rex Ying, Kaidi Cao, Rok Sosič, Ridwan Jalali, Hassan Hamam, Marko Maucec, and Jure Leskovec.
Unravelling the performance of physics-informed graph neural networks for dynamical systems. NIPS, 2022. paper

Abishek Thangamuthu, Gunjan Kumar, Suresh Bishnoi, Ravinder Bhattoo, N M Anoop Krishnan, and Sayan Ranu.
Learning the solution operator of boundary value problems using graph neural networks. ICML, 2022. paper

Winfried Lötzsch, Simon Ohler, and Johannes S. Otterbach.
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Han Gao, Matthew J.Zahr, and Jianxun Wang.
Modular flows: Differential molecular generation. NIPS, 2022. paper

Yogesh Verma, Samuel Kaski, Markus Heinonen, and Vikas Garg.
Learning to solve PDE-constrained inverse problems with graph networks. ICML, 2022. paper

Zhao Qingqing, David B. Lindell, and Gordon Wetzstein.
Physics-embedded neural networks: E(n)-equivariant graph neural PDE solvers. NIPS, 2022. paper

Masanobu Horie and Naoto Mitsume.
PDE-GCN: Novel architectures for graph neural networks motivated by partial differential equations. ICLR, 2021. paper

Moshe Eliasof, Eldad Haber, and Eran Treister.
Physics-aware difference graph networks for sparsely-observed dynamics. ICLR, 2020. paper

Sungyong Seo, Chuizheng Meng, and Yan Liu.
Combining differentiable PDE solvers and graph neural networks for fluid flow prediction. ICML, 2022. paper

Filipe de Avila Belbute-Peres, Thomas D. Economon, and J. Zico Kolter.
Learning continuous-time PDEs from sparse data with graph neural networks. ICLR, 2021. paper

Valerii Iakovlev, Markus Heinonen, and Harri Lähdesmäki.
Learning to simulate complex physics with graph networks. ICML, 2020. paper

Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter W. Battaglia.
Multi-scale physical representations for approximating PDE solutions with graph neural operators. ICLR, 2022. paper

Léon Migus, Yuan Yin, Jocelyn Ahmed Mazari, and Patrick Gallinari.
DS-GPS: A deep statistical graph Poisson solver (for faster CFD simulations). NIPS, 2022. paper

Matthieu Nastorg, Marc Schoenauer, Guillaume Charpiat, Thibault Faney, Jean-Marc Gratien, and Michele-Alessandro Bucci.
GRAND: Graph neural diffusion. ICML, 2021. paper

Benjamin Paul Chamberlain, James Rowbottom, Maria I. Gorinova, Stefan D Webb, Emanuele Rossi, and Michael M. Bronstein.
GRAND++: Graph neural diffusion with a source term. ICML, 2022. paper

Matthew Thorpe, Tan Minh Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher, and Bao Wang.
Neural networks trained to solve differential equations learn general representations. NIPS, 2018. paper

Martin Magill, Faisal Qureshi, and Hendrick de Haan.
Graph element networks: Adaptive, structured computation and memory. ICML, 2019. paper

Ferran Alet, Adarsh Keshav Jeewajee, Maria Bauza Villalonga, Alberto Rodriguez, Tomas Lozano-Perez, and Leslie Kaelbling.
Physics-constrained unsupervised learning of partial differential equations using meshes. arXiv, 2022. paper

Mike Y. Michelis and Robert K. Katzschmann.
Neural PDE solvers for irregular domains. arXiv, 2022. paper

Biswajit Khara, Ethan Herron, Zhanhong Jiang, Aditya Balu, Chih-Hsuan Yang, Kumar Saurabh, Anushrut Jignasu, Soumik Sarkar, Chinmay Hegde, Adarsh Krishnamurthy, and Baskar Ganapathysubramanian.
Bi-stride multi-scale graph neural network for mesh-based physical simulation. arXiv, 2022. paper

Yadi Cao, Menglei Chai, Minchen Li, and Chenfanfu Jiang.
Learning time-dependent PDE solver using message passing graph neural networks. arXiv, 2022. paper

Pourya Pilva and Ahmad Zareei.
On the robustness of graph neural diffusion to topology perturbations. arXiv, 2022. paper

Yang Song, Qiyu Kang, Sijie Wang, Zhao Kai, and Wee Peng Tay.
STONet: A neural-operator-driven spatio-temporal network. arXiv, 2022. paper

Haitao Lin, Guojiang Zhao, Lirong Wu, and Stan Z. Li.
PhyGNNet: Solving spatiotemporal PDEs with physics-informed graph neural network. arXiv, 2022. paper

Longxiang Jiang, Liyuan Wang, Xinkun Chu, Yonghao Xiao, and Hao Zhang.
Differentiable physics-informed graph networks. arXiv, 2019. paper

Sungyong Seo and Yan Liu.
MG-GNN: Multigrid graph neural networks for learning multilevel domain decomposition methods. arXiv, 2023. paper

Ali Taghibakhshi, Nicolas Nytko, Tareq Uz Zaman, Scott MacLachlan, Luke Olson, and Matthew West.
Multi-scale message passing neural PDE solvers. arXiv, 2023. paper

Léonard Equer, T. Konstantin Rusch, and Siddhartha Mishra.
An implicit GNN solver for Poisson-like problems. arXiv, 2023. paper

Matthieu Nastorg, Michele-Alessandro Bucci, Thibault Faney, Jean-Marc Gratien, Guillaume Charpiat, and Marc Schoenauer.
GNN-based physics solver for time-independent PDEs. arXiv, 2023. paper

Rini Jasmine Gladstone, Helia Rahmani, Vishvas Suryakumar, Hadi Meidani, Marta D'Elia, and Ahmad Zareei.
E(3) equivariant graph neural networks for particle-based fluid mechanics. ICLR, 2023. paper

Artur P. Toshev, Gianluca Galletti, Johannes Brandstetter, Stefan Adami, and Nikolaus A. Adams.
Long-short-range message-passing: A physics-informed framework to capture non-local interaction for scalable molecular dynamics simulation. arXiv, 2023. paper

Yunyang Li, Yusong Wang, Lin Huang, Han Yang, Xinran Wei, Jia Zhang, Tong Wang, Zun Wang, Bin Shao, and Tieyan Liu.
A graph convolutional autoencoder approach to model order reduction for parametrized PDEs. arXiv, 2023. paper

Federico Pichi, Beatriz Moya, and Jan S. Hesthaven.

Green Function

Learning Green's functions associated with time-dependent partial differential equations. JMLR, 2022. paper

Nicolas Boullé, Seick Kim, Tianyi Shi, and Alex Townsend.
BI-GreenNet: Learning Green's functions by boundary integral network. arXiv, 2022. paper

Guochang Lin, Fukai Chen, Pipi Hu, Xiang Chen, Junqing Chen, Jun Wang, and Zuoqiang Shi.
DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems. Scientific Reports, 2021. paper

Craig R. Gin, Daniel E. Shea, Steven L. Brunton, and J. Nathan Kutz.
Data-driven discovery of Green’s functions with human-understandable deep learning. Scientific Reports, 2022. paper

Nicolas Boullé, Christopher J. Earls, and Alex Townsend.
Data-driven discovery of Green's functions. Doctoral Dissertation, 2022. Ph.D.

Nicolas Boullé.
Principled interpolation of Green's functions learned from data. arXiv, 2022. paper

Harshwardhan Praveen, Nicolas Boulle, and Christopher Earls.

Finite Element

Composing partial differential equations with physics-aware neural networks. ICML, 2022. paper

Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, and Wolfgang Nowak.
Learning the dynamics of physical systems from sparse observations with finite element networks. ICLR, 2022. paper

Marten Lienen and Stephan Günnemann.
A unified hard-constraint framework for solving geometrically complex PDEs. NIPS, 2022. paper

Songming Liu, Zhongkai Hao, Chengyang Ying, Hang Su, Jun Zhu, and Ze Cheng.
LSH-SMILE: Locality sensitive Hashing accelerated simulation and learning. NIPS, 2021. paper

Chonghao Sima and Yexiang Xue.
Amortized finite element analysis for fast PDE-constrained optimization. ICML, 2020. paper

Tianju Xue, Alex Beatson, Sigrid Adriaenssens, and Ryan Adams.
Learning neural PDE solvers with convergence guarantees. ICLR, 2019. paper

Jun-Ting Hsieh, Shengjia Zhao, Stephan Eismann, Lucia Mirabella, and Stefano Ermon.
Hybrid finite difference with the physics-informed neural network for solving PDE in complex geometries. arXiv, 2022. paper

Zixue Xiang, Wei Peng, Weien Zhou, and Wen Yao.
Multilayer perceptron-based surrogate models for finite element analysis. arXiv, 2022. paper

Lawson Oliveira Lima, Julien Rosenberger, Esteban Antier, and Frederic Magoules.

Convolution

PDE-Net: Learning PDEs from data. ICML, 2018. paper

Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong.
PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. JCP, 2019. paper

Zichao Long, Yiping Lu, and Bin Dong.
Physics-informed CNNs for super-resolution of sparse observations on dynamical systems. NIPS, 2022. paper

Daniel Kelshaw, Georgios Rigas, and Luca Magri.
Deep-pretrained-FWI: Combining supervised learning with physics-informed neural network. NIPS, 2022. paper

Ana Paula O. Muller, Clecio R. Bom, Jessé C. Costa, Matheus Klatt, Elisângela L. Faria, Marcelo P. de Albuquerque, and Márcio P. de Albuquerque.
Learning time-dependent PDEs with a linear and nonlinear separate convolutional neural network. JCP, 2022. paper

Jiagang Qu, Weihua Cai, and YijunZhao.
PhyCRNet: Physics-informed convolutional-recurrent network for solving spatiotemporal PDEs. JCP, 2022. paper

Pu Ren, Chengping Rao, Yang Liu, Jianxun Wang, and Hao Sun.
Spline-PINN: Approaching PDEs without data using fast, physics-informed Hermite-Spline CNNs. AAAI, 2019. paper

Nils Wandel, Michael Weinmann, Michael Neidlin, and Reinhard Klein.
Deep convolutional Ritz method: Parametric PDE surrogates without labeled data. arXiv, 2022. paper

Jan Niklas Fuhg, Arnav Karmarkar, Teeratorn Kadeethum, Hongkyu Yoon, and Nikolaos Bouklas.
Deep convolutional architectures for extrapolative forecasts in time-dependent flow problems. arXiv, 2022. paper

Pratyush Bhatt, Yash Kumar, and Azzeddine Soulaimani.
Phase2Vec: Dynamical systems embedding with a physics-informed convolutional network. arXiv, 2022. paper

Matthew Ricci, Noa Moriel, Zoe Piran, and Mor Nitzan.
Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs. arXiv, 2022. paper

Xu Xiao and Wenlin Qiu.
SplineCNN: Fast geometric deep learning with continuous B-spline kernels. CVPR, 2018. paper

Matthias Fey, Jan Eric Lenssen, Frank Weichert, and Heinrich Muller.
Physics-informed deep super-resolution for spatiotemporal data. arXiv, 2022. paper

Pu Ren, Chengping Rao, Yang Liu, Zihan Ma, Qi Wang, Jianxun Wang, and Hao Sun.
MeshFreeFLowNet: A physics-constrained deep continuous space-time super-resolution framework. SC20, 2020. paper

Chiyu Max Jiang, Soheil Esmaeilzadeh, Kamyar Azizzadenesheli, Karthik Kashinath, Mustafa Mustafa, Hamdi A. Tchelepi, Philip Marcus Prabhat, and Anima Anandkumar.
Convolutional neural operators. arXiv, 2023. paper

Bogdan Raonić, Roberto Molinaro, Tobias Rohner, Siddhartha Mishra, and Emmanuel de Bezenac.
An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator. Computer Methods in Applied Mechanics and Engineering, 2023. paper

Arda Mavi, Ali Can Bekar, Ehsan Haghigh, and Erdogan Madenci.
Clifford neural layers for PDE modeling. ICLR, 2023. paper

Johannes Brandstetter, Rianne van den Berg, Max Welling, and Jayesh K Gupta.
Neural partial differential equations with functional convolution. arXiv, 2023. paper

Ziqian Wu, Xingzhe He, Yijun Li, Cheng Yang, Rui Liu, Shiying Xiong, and Bo Zhu.
Multilevel CNNs for parametric PDEs. arXiv, 2023. paper

Cosmas Heiß, Ingo Gühring, and Martin Eigel.

AutoEncoder

Integral autoencoder network for discretization-invariant learning. JMLR, 2022. paper

Yong Zheng Ong, Zuowei Shen, and Haizhao Yang.
Variational autoencoding neural operators. arXiv, 2023. paper

Jacob H. Seidman, Georgios Kissas, George J. Pappas, and Paris Perdikaris.

Neural Operator

Multiwavelet-based operator learning for differential equations. NIPS, 2021. paper

Gaurav Gupta, Xiongye Xiao, and Paul Bogdan.
On the representation of solutions to elliptic PDEs in Barron space. NIPS, 2021. paper

Ziang Chen, Jianfeng Lu, and Yulong Lu.
Low-rank registration based manifolds for convection-dominated PDEs. AAAI, 2021. paper

Rambod Mojgani and Maciej Balajewicz.
Isogeometric neural networks: A new deep learning approach for solving parameterized partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2023. paper

Joshua Gasick and Xiaoping Qian.
A nonlocal physics-informed deep learning framework using the peridynamic differential operator. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Ehsan Haghighat, Ali CanBekar, Erdogan Madenci, and Ruben Juanes.
Kernel flows: From learning kernels from data into the abyss. JCP, 2019. paper

Houman Owhadi and Gene Ryan Yoo.
On neurosymbolic solutions for PDEs. arXiv, 2022. paper

Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, and Lovekesh Vig.
An introduction to kernel and operator learning methods for homogenization by self-consistent clustering analysis. arXiv, 2022. paper

Owen Huang, Sourav Saha, Jiachen Guo, and Wing Kam Liu.
Nonparametric learning of kernels in nonlocal operators. arXiv, 2022. paper

Fei Lu, Qingci An, and Yue Yu.
Spectral neural operators. arXiv, 2022. paper

V. Fanasko and I. Oseledets.
NOMAD: Nonlinear manifold decoders for operator learning. arXiv, 2022. paper

Jacob H. Seidman, Georgios Kissas, Paris Perdikaris, and George J. Pappas.
U-NO: U-shaped neural operators. arXiv, 2022. paper

Md Ashiqur Rahman, Zachary E. Ross, and Kamyar Azizzadenesheli.
Wavelet neural operator: A neural operator for parametric partial differential equations. arXiv, 2022. paper

Tapas Tripura and Souvik Chakraborty.
Pseudo-differential integral operator for learning solution operators of partial differential equations. arXiv, 2022. paper

Jin Young Shin, Jae Yong Lee, and Hyung Ju Hwang.
Nonlinear reconstruction for operator learning of PDEs with discontinuities. arXiv, 2022. paper

Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn, and Siddhartha Mishra.
GeONet: A neural operator for learning the Wasserstein geodesic. arXiv, 2022. paper

Andrew Gracyk and Xiaohui Chen.
Render unto numerics: Orthogonal polynomial neural operator for PDEs with non-periodic boundary conditions. arXiv, 2022. paper

Ziyuan Liu, Haifeng Wang, Kaijuna Bao, Xu Qian, Hong Zhang, and Songhe Song.
DOSNet as a non-black-box PDE solver: When deep learning meets operator splitting. arXiv, 2022. paper

Yuan Lan, Zhen Li, Jie Sun, and Yang Xiang.
Guiding continuous operator learning through physics-based boundary constraints. arXiv, 2022. paper

Nadim Saad, Gaurav Gupta, Shima Alizadeh, and Danielle C. Maddix.
BelNet: Basis enhanced learning, a mesh-free neural operator. arXiv, 2022. paper

Zecheng Zhang, Wing Tat Leung, and Hayden Schaeffer.
INO: Invariant neural operators for learning complex physical systems with momentum conservation. arXiv, 2022. paper

Ning Liu, Yue Yu, Huaiqian You, and Neeraj Tatikola.
BINN: A deep learning approach for computational mechanics problems based on boundary integral equations. arXiv, 2023. paper

Jia Sun, Yinghua Liu, Yizheng Wang, Zhenhan Yao, and Xiaoping Zheng.
Koopman neural operator as a mesh-free solver of non-linear partial differential equations. arXiv, 2023. paper

Wei Xiong, Xiaomeng Huang, Ziyang Zhang, Ruixuan Deng, Pei Sun, and Yang Tian.
Physics-informed Koopman network. arXiv, 2022. paper

Yuying Liu, Aleksei Sholokhov, Hassan Mansour, and Saleh Nabi.
Deep operator learning lessens the curse of dimensionality for PDEs. arXiv, 2023. paper

Ke Chen, Chunmei Wang, and Haizhao Yang.
Algorithmically designed artificial neural networks (ADANNs): Higher order deep operator learning for parametric partial differential equations. arXiv, 2023. paper

Arnulf Jentzen, Adrian Riekert, and Philippe von Wurstemberger.
Entropy-dissipation informed neural network for McKean-Vlasov type. arXiv, 2023. paper

Zebang Shen and Zhenfu Wang.
A neural PDE solver with temporal stencil modeling. arXiv, 2023. paper

Zhiqing Sun, Yiming Yang, and Shinjae Yoo.
Neural operator learning for long-time integration in dynamical systems with recurrent neural networks. arXiv, 2023. paper

Katarzyna Michałowska, Somdatta Goswami, George Em Karniadakis, and Signe Riemer-Sørensen.
Coupled multiwavelet operator learning for coupled differential equations. ICLR, 2023. paper

Xiongye Xiao, Defu Cao, Ruochen Yang, Gaurav Gupta, Chenzhong Yin, Gengshuo Liu, Radu Balan, and Paul Bogdan.
LNO: Laplace neural operator for solving differential equations. arXiv, 2023. paper

Qianying Cao, Somdatta Goswami, and George Em Karniadakis.

Identification

Data-driven discovery of partial differential equations. SA, 2017. paper

Wei Cai and Zhiqin John Xu.
Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression. NC, 2021. paper

Patrick A. K. Reinbold, Logan M. Kageorge, Michael F. Schatz, and Roman O. Grigoriev.
Discovering nonlinear PDEs from scarce data with physics-encoded learning. ICLR, 2022. paper

Chengping Rao, Pu Ren, Yang Liu, and Hao Sun.
Differential spectral normalization (DSN) for PDE discovery. AAAI, 2021. paper

Chi Chiu So, Tsz On Li, Chufang Wu, and Siu Pang Yung.
Learning differential operators for interpretable time series modeling. KDD, 2022. paper

Yingtao Luo, Chang Xu, Yang Liu, Weiqing Liu, Shun Zheng, and Jiang Bian.
Learning emergent partial differential equations in a learned emergent space. NC, 2022. paper

Felix P. Kemeth, Tom Bertalan, Thomas Thiem, Felix Dietrich, Sung Joon Moon, Carlo R. Laing, and Ioannis G. Kevrekidis.
Physics-informed learning of governing equations from scarce data. NC, 2021. paper

Zhao Chen, Yang Liu, and Hao Sun.
Machine learning hidden symmetries. PRL, 2022. paper

Ziming Liu and Max Tegmark.
Efficient learning of sparse and decomposable PDEs using random projection. UAI, 2022. paper

Md Nasim, Xinghang Zhang, Anter El-Azab, and Yexiang Xue.
Data-driven discovery of partial differential equation models with latent variables. Physical Review E, 2019. paper

Patrick A. K. Reinbold and Roman O. Grigoriev.
Physics constrained learning for data-driven inverse modeling from sparse observations. Physical Review E, 2019. paper

Kailai Xua and Eric Darve.
Deep neural network modeling of unknown partial differential equations in nodal space. JCP, 2022. paper

Zhen Chen, and Victor Churchill, Kailiang Wu, and Dongbin Xiu.
DeepMOD: Deep learning for model discovery in noisy data. JCP, 2021. paper

Gert-Jan Both, Subham Choudhury, Pierre Sens, and Remy Kusters.
Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method. JCP, 2021. paper

Yuntian Chen, Dou Huang, Dongxiao Zhang, Junsheng Zeng, Nanzhe Wang, Haoran Zhang, and Jinyue Yan.
Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data. JCP, 2021. paper

Hao Xua, Dongxiao Zhang, and Nanzhe Wang.
Peridynamics enabled learning partial differential equations. JCP, 2021. paper

Ali C.Bekar and Erdogan Madenci.
Data-driven deep learning of partial differential equations in modal space. JCP, 2020. paper

Kailiang Wu and Dongbin Xiu.
DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm. JCP, 2020. paper

Hao Xu, Haibin Chang, and Dongxiao Zhang.
Learning nonlocal constitutive models with neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Xuhui Zhou, Jiequn Han, and Heng Xiao.
Data-driven identification of parametric partial differential equations. SIAM Journal on Applied Dynamical Systems, 2019. paper

Samuel Rudy, Alessandro Alla, Steven L. Brunton, and J. Nathan Kutz.
PDE-READ: Human-readable partial differential equation discovery using deep learning. Neural Networks, 2022. paper

Robert Stephany and Christopher Earls.
Discover: Deep identification of symbolic open-form PDEs via enhanced reinforcement-learning. arXiv, 2022. paper

Mengge Du, Yuntian Chen, and Dongxiao Zhang.
ModLaNets: Learning generalisable dynamics via modularity and physical inductive bias. ICML, 2022. paper

Yupu Lu, Shijie Lin, Guanqi Chen, and Jia Pan.
Robust discovery of partial differential equations in complex situations. Physical Review Research, 2021. paper

Hao Xu and Dongxiao Zhang.
Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. JCP, 2020. paper

Nicholas Geneva and Nicholas Zabaras.
Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks. Physical Review E, 2022. paper

Danny Lin and Hsiu-Yu Yu.
Data-driven solutions and parameter discovery of the Sasa–Satsuma equation via the physics-informed neural networks method. Physica D, 2022. paper

Haotian Luo, Lei Wang, Yabin Zhang, Gui Lu, Jingjing Su, and Yinchuan Zhao.
Reliable extrapolation of deep neural operators informed by physics or sparse observations. arXiv, 2022. paper

Min Zhu, Handi Zhang, Anran Jiao, George Em Karniadakis, and Lu Lu.
Neural operator with regularity structure for modeling dynamics driven by SPDEs. arXiv, 2022. paper

Peiyan Hu, Qi Meng, Bingguang Chen, Shiqi Gong, Yue Wang, Wei Chen, Rongchan Zhu, Zhiming Ma, and Tieyan Liu.
A PINN approach to symbolic differential operator discovery with sparse data. arXiv, 2022. paper

Lena Podina, Brydon Eastman, and Mohammad Kohandel.
WeakIdent: Weak formulation for identifying differential equations using narrow-fit and trimming. arXiv, 2022. paper

Mengyi Tang, Wenjing Liao, Rachel Kuske, and Sung Ha Kang.
Predicting parametric spatiotemporal dynamics by multi-resolution PDE structure-preserved deep learning. arXiv, 2022. paper

Xinyang Liu, Hao Sun, and Jianxun Wang.
Data-driven modeling of Landau damping by physics-informed neural networks. arXiv, 2022. paper

Yilan Qin, Jiayu Ma, Mingle Jiang, Chuanfei Dong, Haiyang Fu, Liang Wang, Wenjie Cheng, and Yaqiu Jin.
Discovering governing equations in discrete systems using PINNs. arXiv, 2022. paper

Sheikh Saqlain, Wei Zhu, Efstathios G. Charalampidis, and Panayotis G. Kevrekidis.
Physics-guided generative adversarial network to learn physical models. arXiv, 2023. paper

Kazuo Yonekura.
PDE-Learn: Using deep learning to discover partial differential equations from noisy, limited data. arXiv, 2022. paper

Robert Stephany and Christopher Earls.
PiSL: Physics-informed Spline Learning for data-driven identification of nonlinear dynamical systems. MSSP, 2023. paper

Fangzheng Sun, Yang Liu, Qi Wang, Hao Sun.
Learning physical models that can respect conservation laws. arXiv, 2023. paper

Derek Hansen, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, and Michael W. Mahoney.
Data-driven discovery of intrinsic dynamics. NMI, 2022. paper

Daniel Floryan and Michael D. Graham.
Symbolic regression for PDEs using pruned differentiable programs. ICLR, 2023. paper

Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, Lovekesh Vig, and Venkataramana Runkana.
Symbolic physics learner: Discovering governing equations via Monte Carlo tree search. ICLR, 2023. paper

Fangzheng Sun, Yang Liu, Jianxun Wang, Hao Sun.
Learning neural constitutive laws from motion observations for generalizable PDE dynamics. ICML, 2023. paper

Pingchuan Ma, Peter Yichen Chen, Bolei Deng, Joshua B. Tenenbaum, Tao Du, Chuang Gan, Wojciech Matusik.
Learning partial differential equations in reproducing kernel Hilbert spaces. JMLR, 2023. paper

George Stepaniants.
Finite expression methods for discovering physical laws from data. arXiv, 2023. paper

Zhongyi Jiang, Chunmei Wang, and Haizhao Yang.
Physics-informed representation learning for emergent organization in complex dynamical systems. arXiv, 2023. paper

Adam Rupe, Karthik Kashinath, Nalini Kumar, and James P. Crutchfield.

Machine Learning

Machine learning–accelerated computational fluid dynamics. PNAS, 2021. paper

Dmitrii Kochkov, Jamie A. Smith, Ayya Alieva, and Stephan Hoyer.
Learning data-driven discretization for partial differential equations. PNAS, 2019. paper

Yohai Bar-Sinai, Stephan Hoyer, Jason Hickey, and Michael P. Brenner.
Solving high-dimensional partial differential equations using deep learning. PNAS, 2018. paper

Jiequn Han, Arnulf Jentzen, and Weinan E.
Enhancing computational fluid dynamics with machine learning. NCS, 2022. paper

Ricardo Vinuesa and Steven L. Brunton.
Solving high-dimensional parabolic PDEs using the tensor train format. ICML, 2021. paper

Lorenz Richter, Leon Sallandt, and Nikolas Nüsken.
A hybrid reduced basis and machine-learning algorithm for building surrogate models: A first application to electromagnetism. NIPS, 2022. paper

Alejandro Ribés and Ruben Persicot.
Numerically solving parametric families of high-dimensional Kolmogorov partial differential equations via deep learning. NIPS, 2020. paper

Julius Berner, Markus Dablander, and Philipp Grohs.
Nonlocal kernel network (NKN): A stable and resolution-independent deep neural network. JCP, 2022. paper

Huaiqian You, Yue Yu, Marta D'Elia, Tian Gao, and Stewart Silling.
On computing the hyperparameter of extreme learning machines: Algorithm and application to computational PDEs, and comparison with classical and high-order finite elements. JCP, 2022. paper

Suchuan Dong and Jielin Yang.
Int-Deep: A deep learning initialized iterative method for nonlinear problems. JCP, 2022. paper

Jianguo Huang, Haoqin Wang, and Haizhao Yang.
DeLISA: Deep learning based iteration scheme approximation for solving PDEs. JCP, 2022. paper

Ying Li, Zuojia Zhou, and Shihui Ying.
Physics-informed machine learning for reduced-order modeling of nonlinear problems. JCP, 2021. paper

Wenqian Chen, Qian Wang, Jan S.Hesthaven, and Chuhua Zhang.
SelectNet: Self-paced learning for high-dimensional partial differential equations. JCP, 2021. paper

Yiqi Gu, Haizhao Yang, and Chao Zhou.
Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs. JCP, 2020. paper

Zhiqiang Cai, Jingshuang Chen, Min Liu, and Xinyu Liu.
Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Suchuan Dong and Zongwei Li.
Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Kjetil O.Lye, Siddhartha Mishra, Deep Ray, and Praveen Chandrashekar.
Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets. Computer Methods in Applied Mechanics and Engineering, 2021. paper

C. Soize and R. Ghanem.
An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020. paper

E. Samaniego, C. Anitescu, S.Goswami, V.M.Nguyen-Thanh, H.Guo, K.Hamdia, X.Zhuang, and T.Rabczuk.
Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning. JCP, 2021. paper

Nikolaj T.Mücke, Sander M.Bohté, and Cornelis W.Oosterlee.
GCN-FFNN: A two-stream deep model for learning solution to partial differential equations. Neurocomputing, 2022. paper

Onur Bilgin, Thomas Vergutz, Siamak Mehrkanoon.
Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs. Journal of Scientific Computing, 2021. paper

Ludovica Cicci, Stefania Fresca, and Andrea Manzoni.
HybridNet: Integrating Model-based and Data-driven Learning to Predict Evolution of Dynamical Systems. CoRL, 2018. paper

Yun Long, Xueyuan She, and Saibal Mukhopadhyay.
A model-constrained tangent slope learning approach for dynamical systems. arXiv, 2022. paper

Hai V. Nguyen and Tan Bui-Thanh.
Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations. arXiv, 2022. paper

Fengjiang Fu and Xiaoqun Wang.
Solving PDEs on unknown manifolds with machine learning. arXiv, 2021. paper

Senwei Liang, Shixiao W. Jiang, John Harlim, and Haizhao Yang.
Model reduction and neural networks for parametric PDEs. SIAM Journal of Computational Mathematics, 2021. paper

Kaushik Bhattacharya, Bamdad Hosseini, Nikola B. Kovachki, and Andrew M. Stuart.
MOD-Net: A machine learning approach via model-operator-data network for solving PDEs. Communications in Computational Physics, 2022. paper

Shamsulhaq Basir and Inanc Senocak.
A Deep Double Ritz Method (D^2RM) for solving Partial Differential Equations using Neural Networks. Computer Methods in Applied Mechanics and Engineering, 2023. paper

Carlos Uriarte, David Pardo, Ignacio Muga, and Judit Muñoz-Matute.
Data- and theory-guided learning of partial differential equations using SimultaNeous basis function Approximation and Parameter Estimation (SNAPE). MSSP, 2023. paper

Sutanu Bhowmick, Satish Nagarajaiah, and Anastasios Kyrillidis.
Invariant preservation in machine learned PDE solvers via error correction. arXiv, 2023. paper

Nick McGreivy and Ammar Hakim.
Operator learning with PCA-Net: upper and lower complexity bounds. arXiv, 2023. paper

Samuel Lanthaler.
Multiscale attention via wavelet neural operators for vision Transformers. arXiv, 2023. paper

Anahita Nekoozadeh, Mohammad Reza Ahmadzadeh, Zahra Mardani, and Morteza Mardani.
Pseudo-Hamiltonian neural networks for learning partial differential equations. arXiv, 2023. paper

Sølve Eidnes and Kjetil Olsen Lye.
SHoP: A deep learning framework for solving high-order partial differential equations. arXiv, 2023. paper

Tingxiong Xiao, Runzhao Yang, Yuxiao Cheng, Jinli Suo, and Qionghai Dai.

Neural ODE

Neural rough differential equations for long time series. ICML, 2021. paper

James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster.
LEADS: Learning dynamical systems that generalize across environments. NIPS, 2021. paper

Yuan Yin, Ibrahim Ayed, Emmanuel de Bézenac, and Nicolas Baskiotis, and Patrick Gallinari.
A neural ODE interpretation of transformer layers. NIPS, 2022. paper

Yuan Yin, Ibrahim Ayed, Emmanuel de Bézenac, Nicolas Baskiotis, and Patrick Gallinari.
On robustness of neural ordinary differential equations. ICLR, 2020. paper

Hanshu Yan, Jiawei Du, Vincent Y. F. Tan, and Jiashi Feng.
Spectrum dependent learning curves in kernel regression and wide neural networks. ICML, 2020. paper

Blake Bordelon, Abdulkadir Canatar, and Cengiz Pehlevan.
Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. ICML, 2018. paper

Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong.
Neural ordinary differential equations. NIPS, 2018. paper

Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud.
Hamiltonian neural networks. ICML, 2018. paper

Sam Greydanus, Misko Dzamba, and Jason Yosinski.
Hamiltonian neural networks for solving equations of motion. Physical Review E, 2022. paper

Marios Mattheaki, David Sondak, Akshunna S. Dogra, and Pavlos Protopapas.
Deep learning for universal linear embeddings of nonlinear dynamics. NC, 2018. paper

Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton.
Stochastic physics-informed neural ordinary differential equations. arXiv, 2021. paper

Jared O'Leary, Joel A. Paulson, and Ali Mesbah.
A stable and scalable method for solving initial value PDEs with neural networks. ICLR, 2023. paper

Marc Anton Finzi, Andres Potapczynski, Matthew Choptuik, and Andrew Gordon Wilson.
Characteristic neural ordinary differential equation. ICLR, 2023. paper

Xingzi Xu, Ali Hasan, Khalil Elkhalil, Jie Ding, and Vahid Tarokh.
Efficient certified training and robustness verification of neural ODEs. ICLR, 2023. paper

Mustafa Zeqiri, Mark Niklas Mueller, Marc Fischer, and Martin Vechev.
Efficient certified training and robustness verification of neural ODEs. ICLR, 2023. paper

Mustafa Zeqiri, Mark Niklas Mueller, Marc Fischer, and Martin Vechev.

Mechanism

Library

PDEBench: An extensive benchmark for scientific machine learning. NIPS, 2022. paper

Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Dan MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert.
DeepXDE: A deep learning library for solving differential equations. SIAM Review, 2021. paper

Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis.
A research framework for writing differentiable PDE discretizations in JAX. NIPS, 2021. paper

Antonio Stanziola, Simon R. Arridge, Ben T. Cox, and Bradley E. Treeby.
An extensible benchmarking graph-mesh dataset for studying steady-state incompressible Navier-Stokes equations. ICLR, 2022. paper

Florent Bonnet, Jocelyn Ahmed Mazari, Thibaut Munzer, Pierre Yser, and Patrick Gallinari.
PhiFlow: A differentiable PDE solving framework for deep learning via physical simulations. NIPS, 2020. paper

Philipp Holl, Vladlen Koltun, Kiwon Um, and Nils Thuerey.
NVIDIA SimNet™: An AI-accelerated multi-physics simulation framework. ICCS, 2021. paper

Oliver Hennigh, Susheela Narasimhan, Mohammad Amin Nabian, Akshay Subramaniam, Kaustubh Tangsali, Zhiwei Fang, Max Rietmann, Wonmin Byeon, and Sanjay Choudhry.
PyDEns: A python framework for solving differential equations with neural networks. arXiv, 2019. paper

Alexander Koryagin, Roman Khudorozkov, and Sergey Tsimfer.
KoopmanLab: A PyTorch module of Koopman neural operator family for solving partial differential equations. arXiv, 2023. paper

Wei Xiong, Muyuan Ma, Pei Sun, and Yang Tian.
Physics-driven machine learning models coupling PyTorch and Firedrake. arXiv, 2023. paper

Nacime Bouziani and David A. Ham.

Analysis

Characterizing possible failure modes in physics-informed neural networks. NIPS, 2021. paper

Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W. Mahoney.
Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 2021. paper

Sifan Wang, Yujun Teng, and Paris Perdikaris.
When and why PINNs fail to train: A neural tangent kernel perspective. JCP, 2022. paper

SifanWang, XinlingYu, and Paris Perdikaris.
When do extended physics-informed neural networks (XPINNs) improve generalization? SIAM Journal on Scientific Computing, 2022. paper

Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, and Kenji Kawaguchi.
MARS: A method for the adaptive removal of stiffness in PDEs. JCP, 2022. paper

Laurent Duchemin and Jen Eggers.
Learning similarity metrics for numerical simulations. ICML, 2020. paper

Georg Kohl, Kiwon Um, and Nils Thuerey.
A curriculum-training-based strategy for distributing collocation points during physics-informed neural network training. NIPS, 2021. paper

Marcus Münzer and Chris Bard.
Parametric complexity bounds for approximating PDEs with neural networks. NIPS, 2021. paper

Tanya Marwah, Zachary Lipton, and Andrej Risteski.
Respecting causality is all you need for training physics-informed neural networks. arXiv, 2021. paper

Sifan Wang, Shyam Sankaran, and Paris Perdikaris.
CP-PINNS: Changepoints detection in PDEs using physics informed neural networks with total-variation penalty. arXiv, 2022. paper

Zhikang Dong and Pawel Polak.
Stochastic projection based approach for gradient free physics informed learning. arXiv, 2022. paper

Navaneeth N and Souvik Chakraborty.
Understanding the difficulty of training physics-informed neural networks on dynamical systems. arXiv, 2022. paper

Franz M. Rohrhofer, Stefan Posch, Clemens Gößnitzer, and Bernhard C. Geiger.
Mitigating learning complexity in physics and equality constrained artificial neural networks. arXiv, 2022. paper

Victor Churchill and Dongbin Xiu.
Improved training of physics-informed neural networks with model ensembles. arXiv, 2022. paper

Katsiaryna Haitsiukevich and Alexander Ilin.
The cost-accuracy trade-off in operator learning with neural networks. arXiv, 2022. paper

Maarten V. de Hoop, Daniel Zhengyu Huang, Elizabeth Qian, and Andrew M. Stuart.
Neural tangent kernel analysis of PINN for advection-diffusion equation. arXiv, 2022. paper

M. H. Saadat, B. Gjorgiev, L. Das, and G. Sansavini.
Separable PINN: Mitigating the curse of dimensionality in physics-informed neural networks. NIPS, 2022. paper

Junwoo Cho, Seungtae Nam, Hyunmo Yang, Seok-Bae Yun, Youngjoon Hong, and Eunbyung Park.
A curriculum-training-based strategy for distributing collocation points during physics-informed neural network training. NIPS, 2022. paper

Marcus Münzer and Chris Bard.
Robustness of physics-informed neural networks to noise in sensor data. arXiv, 2022. paper

Jian Cheng Wong, Pao-Hsiung Chiu, Chin Chun Ooi, and My Ha Da.
Investigations on convergence behaviour of physics informed neural networks across spectral ranges and derivative orders. arXiv, 2023. paper

Mayank Deshpande, Siddharth Agarwal, Vukka Snigdha, and Arya Kumar Bhattacharya.
Stochastic projection based approach for gradient free physics informed learning. Computer Methods in Applied Mechanics and Engineering, 2023. paper

Navaneeth N. and Souvik Chakraborty.
Temporal consistency loss for physics-informed neural networks. arXiv, 2023. paper

Sukirt Thakur, Maziar Raissi, Harsa Mitra, and Arezoo Ardekani.
Can physics-informed neural networks beat the finite element method? arXiv, 2023. paper

Tamara G. Grossmann, Urszula Julia Komorowska, Jonas Latz, and Carola-Bibiane Schönlieb.
LSA-PINN: Linear boundary connectivity loss for solving PDEs on complex geometry. arXiv, 2023. paper

Jian Cheng Wong, Pao-Hsiung Chiu, Chinchun Ooi, and My Ha Dao, and Yew-Soon Ong.
On the Hyperparameters influencing a PINN’s generalization beyond the training domain. arXiv, 2023. paper

Andrea Bonfanti, Roberto Santana, Marco Ellero, and Babak Gholami.
DPM: A novel training method for physics-informed neural networks in extrapolation. AAAI, 2021. paper

Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jhin, and Noseong Park.
Guiding continuous operator learning through physics-based boundary constraints. ICLR, 2023. paper

Nadim Saad, Gaurav Gupta, Shima Alizadeh, and Danielle C. Maddix.
A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions. arXiv, 2023. paper

Michael Penwarden, Ameya D. Jagtap, Shandian Zhe, George Em Karniadakis, and Robert M. Kirby.
Machine learning for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality. ICLR, 2022. paper

Yiping Lu, Haoxuan Chen, Jianfeng Lu, Lexing Ying, and Jose Blanchet.
Probing optimisation in physics-informed neural networks. ICLR, 2023. paper

Nayara Fonseca, Veronica Guidetti, and Will Trojak.
Nearly optimal VC-dimension and Pseudo-dimension bounds for deep neural network derivatives. arXiv, 2023. paper

Yahong Yang, Haizhao Yang, and Yang Xiang.
PDE+: Enhancing generalization via PDE with adaptive distributional diffusion. arXiv, 2023. paper

Yige Yuan, Bingbing Xu, Bo Lin, Liang Hou, Fei Sun, Huawei Shen, and Xueqi Cheng.

Attention

Learning operators with coupled attention. JMLR, 2022. paper

Matthew Thorpe, Tan Minh Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher, and Bao Wang.
Choose a Transformer: Fourier or Galerkin. NIPS, 2021. paper

Shuhao Cao.
Predicting physics in mesh-reduced space with temporal attention. ICLR, 2022. paper

Xu Han, Han Gao, Tobias Pfaff, Jianxun Wang, and Liping Liu.
SiT: Simulation transformer for particle-based physics simulation. ICLR, 2022. paper

Yidi Shao, Chen Change Loy, and Bo Dai.
Physics-informed long-sequence forecasting from multi-resolution spatiotemporal data. IJCAI, 2022. paper

Chuizheng Meng, Hao Niu, Guillaume Habault, Roberto Legaspi, Shinya Wada, Chihiro Ono, and Yan Liu.
TransFlowNet: A physics-constrained Transformer framework for spatio-temporal super-resolution of flow simulations. Journal of Computational Science, 2022. paper

Xinjie Wang, Siyuan Zhu, Yundong Guo, Peng Han, Yucheng Wang, Zhiqiang Wei, and Xiaogang Jin.
Self-adaptive physics-informed neural networks using a soft attention mechanism. AAAI-MLPS, 2021. paper

Levi D. McClenny and Ulisses Braga-Neto.
Transformer for partial differential equations' operator learning. arXiv, 2022. paper

Zijie Li, Kazem Meidani, and Amir Barati Farimani.
Physics-informed attention-based neural network for hyperbolic partial differential equations: Application to the Buckley–Leverett problem. Scientific Reports, 2022. paper

Ruben Rodriguez-Torrado, Pablo Ruiz, Luis Cueto-Felgueroso, Michael Cerny Green, Tyler Friesen, Sebastien Matringe, and Julian Togelius.
Self-adaptive physics-informed neural networks using a soft attention mechanism. AAAI-MLPS, 2021. paper

Levi Mc Clenny and Ulisses Braga-Neto.
Nonlinear reconstruction for operator learning of PDEs with discontinuities. ICLR, 2023. paper

Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn, and Siddhartha Mishra.
GNOT: A general neural operator transformer for operator learning. arXiv, 2023. paper

Zhongkai Hao, Chengyang Ying, Zhengyi Wang, Hang Su, Yinpeng Dong, Songming Liu, Ze Cheng, Jun Zhu, and Jian Song.
In-context operator learning for differential equation problems. arXiv, 2023. paper

Liu Yang, Siting Liu, Tingwei Meng, and Stanley J. Osher.
Learning neural PDE solvers with parameter-guided channel attention. ICML, 2023. paper

Makoto Takamoto, Francesco Alesiani, and Mathias Niepert.
Physics informed token Transformer. arXiv, 2023. paper

Cooper Lorsung, Zijie Li, and Amir Barati Farimani.

Neural Implicit Flow

Neural implicit flow: A mesh-agnostic dimensionality reduction paradigm of spatio-temporal data. arXiv, 2022. paper

Shaowu Pan, Steven L. Brunton, and J. Nathan Kutz.
CROM: Continuous reduced-order modeling of PDEs using implicit neural representations. ICLR, 2023. paper

Peter Yichen Chen, Jinxu Xiang, Dong Heon Cho, Yue Chang, G A Pershing, Henrique Teles Maia, Maurizio Chiaramonte, Kevin Carlberg, and Eitan Grinspun.
MAgNet: Mesh agnostic neural PDE solver. NIPS, 2022. paper

Oussama Boussif, Dan Assouline, Loubna Benabbou, and Yoshua Bengio.
NTopo: Mesh-free topology optimization using implicit neural representations. NIPS, 2021. paper

Jonas Zehnder, Yue Li, Stelian Coros, and Bernhard Thomaszewski.
ContactNets: Learning discontinuous contact dynamics with smooth, implicit representations. ICLR, 2020. paper

Samuel Pfrommer, Mathew Halm, and Michael Posa.
Continuous PDE dynamics forecasting with implicit neural representations. ICLR, 2023. paper

Yuan Yin, Matthieu Kirchmeyer, Jean-Yves Franceschi, Alain Rakotomamonjy, and Patrick Gallinari.

Disentangle

PDE-driven spatiotemporal disentanglement. ICLR, 2021. paper

Jérémie Donà, Jean-Yves Franceschi, Sylvain Lamprier, and Patrick Gallinari.
Disentangling physical dynamics from unknown factors for unsupervised video prediction. CVPR, 2020. paper

Vincent Le Guen and Nicolas Thome.

Meta Learning

Meta-auto-decoder for solving parametric partial differential equations. NIPS, 2022. paper

Xiang Huang, Zhanhong Ye, Hongsheng Liu, Beiji Shi, Zidong Wang, Kang Yang, Yang Li, Bingya Weng, Min Wang, Haotian Chu, Jing Zhou, Fan Yu, Bei Hua, Lei Chen, and Bin Dong.
Transfer learning with physics-informed neural networks for efficient simulation of branched flows. NIPS, 2022. paper

Raphaël Pellegrin, Blake Bullwinkel, Marios Mattheakis, and Pavlos Protopapas.
Identifying physical law of hamiltonian systems via meta-learning. ICLR, 2021. paper

Seungjun Lee, Haesang Yang, and Woojae Seong.
One-shot learning for solution operators of partial differential equations. ICLR, 2021. paper

Lu Lu, Haiyang He, Priya Kasimbeg, Rishikesh Ranade, and Jay Pathak.
A metalearning approach for physics-informed neural networks (PINNs): Application to parameterized PDEs. JCP, 2023. paper

Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M.Kirby.
Meta-MgNet: Meta multigrid networks for solving parameterized partial differential equations. JCP, 2022. paper

Yuyan Chen, Bin Dong, and Jinchao Xu.
Mosaic flows: A transferable deep learning framework for solving PDEs on unseen domains. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Hengjie Wang, Robert Planas, Aparna Chandramowlishwaran, and Ramin Bostanabad.
Meta-learning PINN loss functions. JCP, 2022. paper

Apostolos F Psaros,Kenji Kawaguchi, and George Em Karniadakis.
HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks. NIPS, 2021. paper

Filipe de Avila Belbute-Peres, Yifan Chen, and Fei Sha.
Adversarial multi-task learning enhanced physics-informed neural networks for solving partial differential equations. IJCNN, 2022. paper

Pongpisit Thanasutives, Masayuki Numao, and Ken-ichi Fukui.
Transfer physics informed neural network: A new framework for distributed physics informed neural networks via parameter sharing. Engineering with Computers, 2022. paper

Sreehari Manikkan and Balaji Srinivasan.
Meta-PDE: Learning to solve PDEs quickly without a mesh. arXiv, 2022. paper

Tian Qin, Alex Beatson, Deniz Oktay, Nick Mc Greivy, and Ryan P. Adams.
SVD-PINNs: Transfer learning of physics-informed neural networks via singular value decomposition. arXiv, 2022. paper

Yihang Gao, Ka Chun Cheung, and Michael K. Ng.
Physics-informed neural networks (PINNs) for parameterized PDEs: A meta-learning approach. arXiv, 2021. paper

Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M. Kirby.
Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs. arXiv, 2022. paper

Anastasia Borovykh, Dante Kalise, Alexis Laignelet, and Panos Parpas.
Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios. arXiv, 2022. paper

Chen Xu, Ba Trung Cao, Yong Yuan, and Günther Meschke.
TransNet: Transferable neural networks for partial differential equations. arXiv, 2023. paper

Zezhong Zhang, Feng Bao, Lili Ju, and Guannan Zhang.
Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems. arXiv, 2023. paper

Sarah Perez, Suryanarayana Maddu, Ivo F. Sbalzarini, and Philippe Poncet.
GPT-PINN: Generative pre-trained physics-informed neural networks toward non-intrusive meta-learning of parametric PDEs. arXiv, 2023. paper

Yanlai Chen and Shawn Koohy.
A metalearning approach for physics-informed neural networks (PINNs): Application to parameterized PDEs. JCP, 2023. paper

Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M. Kirby.
Gradient-enhanced physics-informed neural networks based on transfer learning for inverse problems of the variable coefficient differential equations. arXiv, 2023. paper

Shuning Lin and Yong Chen.

AutoML

Auto-PINN: Understanding and optimizing physics-informed neural architecture. arXiv, 2022. paper

Yicheng Wang, Xiaotian Han, Chiayuan Chang, Daochen Zha, Ulisses Braga-Neto, and Xia Hu.
Building high accuracy emulators for scientific simulations with deep neural architecture search. Machine Learning: Science and Technology, 2022. paper

M F Kasim, D Watson-Parris, L Deaconu, S Oliver, P Hatfield, D H Froula, G Gregori, M Jarvis, S Khatiwala, J Korenaga, J Topp-Mugglestone, E Viezzer, and S M Vinko.
Learning operations for neural PDE solvers. ICLR, 2019. paper

Nicholas Roberts, Mikhail Khodak, Tri Dao, Liam Li, Christopher Re, and Ameet Talwalkar.
AutoPINN: When AutoML meets physics-informed neural networks. arXiv, 2022. paper

Xinle Wu, Dalin Zhang, Miao Zhang, Chenjuan Guo, Shuai Zhao, Yi Zhang, Huai Wang, and Bin Yang.
NAS-PINN: Neural architecture search-guided physics-informed neural network for solving PDEs. arXiv, 2023. paper

Yifan Wang and Linlin Zhong.

Sampling

ADLGM: An efficient adaptive sampling deep learning Galerkin method. JCP, 2023. paper

Andreas C.Aristotelous, Edward C.Mitchell, and Vasileios Maroulasb.
DMIS: Dynamic mesh-based importance sampling for training physics-informed neural networks. AAAI, 2023. paper

Zijiang Yang, Zhongwei Qiu, and Dongmei Fu.
A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Chenxi Wua, Min Zhua, Qinyang Tan, YadhKartha, and Lu Lu.
Mitigating propagation failures in PINNs using evolutionary sampling. arXiv, 2022. paper

Arka Daw, Jie Bu, Sifan Wang, Paris Perdikaris, and Anuj Karpatne.
Residual-quantile adjustment for adaptive training of physics-informed neural network. arXiv, 2022. paper

Jiayue Han, Zhiqiang Cai, Zhiyou Wu, and Xiang Zhou.
Adversarial sampling for solving differential equations with neural networks. NIPS, 2021. paper

Kshitij Parwani and Pavlos Protopapas.
A novel adaptive causal sampling method for physics-informed neural networks. arXiv, 2022. paper

Jia Guo, Haifeng Wang, and Chenping Hou.
Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions. arXiv, 2023. paper

Zhiping Mao and Xuhui Meng.
Active learning based sampling for high-dimensional nonlinear partial differential equations. JCP, 2023. paper

Wenhan Gao and Chunmei Wang.

Latent Space

Multiscale simulations of complex systems by learning their effective dynamics. NMI, 2022. paper

Pantelis R. Vlachas, Georgios Arampatzis, Caroline Uhler, and Petros Koumoutsakos.
A latent space solver for PDE generalization. ICLR, 2021. paper

Rishikesh Ranade, Chris Hill, Haiyang He, Amir Maleki, and Jay Pathak.
Approximate latent force model inference. AAAI, 2021. paper

Jacob D. Moss, Felix L. Opolka, Bianca Dumitrascu, and Pietro Lió.
Learning to accelerate partial differential equations via latent global evolution. NIPS, 2022. paper

Tailin Wu, Takashi Maruyama, and Jure Leskovec.
Exploring physical latent spaces for deep learning. arXiv, 2022. paper

Chloe Paliard, Nils Thuerey, and Kiwon Um.
Certified data-driven physics-informed greedy auto-encoder simulator. arXiv, 2022. paper

Xiaolong He, Youngsoo Choi, William D. Fries, Jonathan L. Belof, and Jiun-Shyan Chen.
Deep latent regularity network for modeling stochastic partial differential equations. AAAI, 2023. paper

Shiqi Gong, Peiyan Hu, Qi Meng, Yue Wang, Rongchan Zhu, Bingguang Chen, Zhiming Ma, Hao Ni, and Tieyan Liu.
Learning in latent spaces improves the predictive accuracy of deep neural operators. arXiv, 2023. paper

Katiana Kontolati, Somdatta Goswami, George Em Karniadakis, and Michael D. Shields.

Loss Function

Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. JCP, 2021. paper

Ameya D.Jagtap, KenjiKawaguchi, and George Em Karniadakis.
Adaptive deep neural networks methods for high-dimensional partial differential equations. JCP, 2022. paper

Shaojie Zeng, Zong Zhang, and Qingsong Zou.
Self-adaptive loss balanced Physics-informed neural networks. Neurocomputing, 2022. paper

Zixue Xiang, Wei Peng, Xu Liu, and Wen Yao.
Multi-objective loss balancing for physics-informed deep learning. arXiv, 2021. paper

Rafael Bischof and Michael Kraus.
Self-scalable Tanh (Stan): Faster convergence and better generalization in physics-informed neural networks. arXiv, 2022. paper

Raghav Gnanasambandam, Bo Shen, Jihoon Chung, Xubo Yue, and Zhenyu (James) Kong.
Is L2 physics-informed loss always suitable for training physics-informed neural network? arXiv, 2022. paper

Chuwei Wang, Shanda Li, Di He, and Liwei Wang.
Loss landscape engineering via data regulation on PINNs. arXiv, 2022. paper

Vignesh Gopakumar, Stanislas Pamela, and Debasmita Samaddar.
Implicit stochastic gradient descent for training physics-informed neural networks. AAAI, 2023. paper

Ye Li, Songcan Chen, and Shengjun Huang.
About optimal loss function for training physics-informed neural networks under respecting causality. arXiv, 2023. paper

Vasiliy A. Es'kin, Danil V. Davydov, Ekaterina D. Egorova, Alexey O. Malkhanov, Mikhail A. Akhukov, and Mikhail E. Smorkalov.

Decomposition

Composing partial differential equations with physics-aware neural networks. ICLR, 2022. paper

Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, Wolfgang Nowak, and Martin V. Butz.
Learning composable energy surrogates for PDE order reduction. NIPS, 2020. paper

Alex Beatson, Jordan T. Ash, Geoffrey Roeder, Tianju Xue, and Ryan P. Adams.
Neural basis functions for accelerating solutions to high mach euler equations. ICML, 2022. paper

David Witman, Alexander New, Hicham Alkandry, and Honest Mrema.
PFNN-2: A domain decomposed penalty-free neural network method for solving partial differential equations. arXiv, 2022. paper

Hailong Shen and Chao Yang.
Finite basis physics-informed neural networks (FBPINNs): A scalable domain decomposition approach for solving differential equations. arXiv, 2021. paper

Ben Moseley, Andrew Markham, and Tarje Nissen-Meyer.
Finite basis physics-informed neural networks as a Schwarz domain decomposition method. arXiv, 2022. paper

Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley.
NeuralStagger: Accelerating physics constrained neural PDE solver with spatialtemporal decomposition. arXiv, 2023. paper

Xinquan Huang, Wenlei Shi, Qi Meng, Yue Wang, Xiaotian Gao, Jia Zhang, and Tieyan Liu.
Augmenting physical models with deep networksfor complex dynamics forecasting. Journal of Statistical Mechanics: Theory and Experiment, 2021. paper

Yuan Yin, Vincent Le Guen, Jérémie Dona, Emmanuel de Bézenac, Ibrahim Ayed, Nicolas Thome, and Patrick Gallinari.
LordNet: Learning to solve parametric partial differential equations without simulated data. arXiv, 2022. paper

Wenlei Shi, Xinquan Huang, Xiaotian Gao, Xinran Wei, Jia Zhang, Jiang Bian, Mao Yang, and Tieyan Liu.
An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations. arXiv, 2022. paper

Ivan Prusak, Monica Nonino, Davide Torlo, Francesco Ballarin, and Gianluigi Rozza.
Physics-informed spectral learning: the Discrete Helmholtz-Hodge decomposition. arXiv, 2023. paper

Luis Espath, Pouria Behnoudfar, and Raul Tempone.

Mesh

MeshingNet: A new mesh generation method based on deep learning. ICCS, 2022. paper

Zheyan Zhang, Yongxing Wang, Peter K. Jimack, and He Wang.
M2N: Mesh movement networks for PDE solvers. arXiv, 2022. paper

Wenbin Song, Mingrui Zhang, Joseph G. Wallwork, Junpeng Gao, Zheng Tian, Fanglei Sun, Matthew D. Piggott, Junqing Chen, Zuoqiang Shi, Xiang Chen, and Jun Wang.
RANG: A residual-based adaptive node generation method for physics-informed neural networks. arXiv, 2022. paper

Wei Peng, Weien Zhou, Xiaoya Zhang, Wen Yao, and Zheliang Liu.
Learning a mesh motion technique with application to fluid-structure interaction and shape optimization. arXiv, 2022. paper

Johannes Haubne and Miroslav Kuchta.
Accelerated training of physics-informed neural networks (PINNs) using meshless discretizations. arXiv, 2022. paper

Ramansh Sharma and Varun Shankar.
An improved structured mesh generation method based on physics-informed neural networks. arXiv, 2022. paper

Xinhai Chen, Jie Liu, Junjun Yan, Zhichao Wang, and Chunye Gong.
Mesh-free Eulerian physics-informed neural networks. arXiv, 2022. paper

Fabricio Arend Torres, Marcello Massimo Negri, Monika Nagy-Huber, Maxim Samarin, and Volker Roth.
Fixed-budget online adaptive mesh learning for physics-informed neural networks. Towards parameterized problem inference. arXiv, 2022. paper

Thi Nguyen Khoa Nguyen, Thibault Dairay, Raphaël Meunier, Christophe Millet, and Mathilde Mougeot.
Learning controllable adaptive simulation for multi-resolution physics. ICLR, 2023. paper

Tailin Wu, Takashi Maruyama, Qingqing Zhao, Gordon Wetzstein, and Jure Leskovec.
A closest point method for surface PDEs with interior boundary conditions for geometry processing. arXiv, 2023. paper

Nathan King, Haozhe Su, Mridul Aanjaneya, Steven Ruuth, and Christopher Batty.

Generative Model

A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks. NCS, 2021. paper

Teeratorn Kadeethum, Daniel O’Malley, Jan Niklas Fuhg, Youngsoo Choi, Jonghyun Lee, Hari S. Viswanathan, and Nikolaos Bouklas.
Neural SDEs as infinite-dimensional GANs. ICML, 2021. paper

Patrick Kidger, James Foster, Xuechen Li, and Terry J Lyons.
PID-GAN: A GAN framework based on a physics-informed discriminator for uncertainty quantification with physics. KDD, 2021. paper

Arka Daw, M. Maruf, and Anuj Karpatne.
Generative adversarial neural operators. Transactions on Machine Learning Research, 2022. paper

Md Ashiqur Rahman, Manuel A Florez, Anima Anandkumar, Zachary E Ross, and Kamyar Azizzadenesheli.
Weak adversarial networks for high-dimensional partial differential equations. JCP, 2020. paper

Yaohua Zang, Gang Bao, Xiaojing Ye, and Haomin Zhou.
Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks. JCP, 2022. paper

Yihang Gao and Michael K. Ng.
Competitive physics informed networks. ICLR, 2023. paper

Qi Zeng, Yash Kothari, Spencer H Bryngelson, and Florian Tobias Schaefer.
Learning generative neural networks with physics knowledge. Research in the Mathematical Sciences, 2022. paper

Kailai Xu, Weiqiang Zhu, and Eric Darve.
Diffusion generative models in infinite dimensions. arXiv, 2022. paper

Gavin Kerrigan, Justin Ley, and Padhraic Smyth.
Revisiting PINNs: Generative adversarial physics-informed neural networks and point-weighting method. arXiv, 2022. paper

Wensheng Li, Chao Zhang, Chuncheng Wang, Hanting Guan, and Dacheng Tao.
PIAT: Physics informed adversarial training for solving partial differential equations. arXiv, 2022. paper

Simin Shekarpaz, Mohammad Azizmalayeri, and Mohammad Hossein Rohban.
Physics-constrained generative adversarial networks for 3D turbulence. arXiv, 2022. paper

Dima Tretiak, Arvind T. Mohan, and Daniel Livescu.
Score-based diffusion models in function space. arXiv, 2023. paper

Jae Hyun Lim, Nikola B. Kovachki, Ricardo Baptista, Christopher Beckham, Kamyar Azizzadenesheli, Jean Kossaifi, Vikram Voleti, Jiaming Song, Karsten Kreis, Jan Kautz, Christopher Pal, Arash Vahdat, and Anima Anandkumar.
Infinite-dimensional diffusion models for function spaces. arXiv, 2023. paper

Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang.
A physics-informed diffusion model for high-fidelity flow field reconstruction. JCP, 2023. paper

Dule Shu, Zijie Li, and Amir Barati Farimani.
Generative modelling with inverse heat dissipation. ICLR, 2023. paper

Severi Rissanen, Markus Heinonen, and Arno Solin.
Differentiable Gaussianization layers for inverse problems regularized by deep generative models. ICLR, 2023. paper

Dongzhuo Li.
Transformer meets boundary value inverse problems. ICLR, 2023. paper

Ruchi Guo, Shuhao Cao, and Long Chen.
ViTO: Vision Transformer-operator. arXiv, 2023. paper

Oded Ovadia, Adar Kahana, Panos Stinis, Eli Turkel, and George Em Karniadakis.
LatentPINNs: Generative physics-informed neural networks via a latent representation learning. arXiv, 2023. paper

Mohammad H. Taufik and Tariq Alkhalifah.
Generative diffusion learning for parametric partial differential equations. arXiv, 2023. paper

Ting Wang, Petr Plechac, and Jaroslaw Knap.
Scalable Transformer for PDE surrogate modeling. arXiv, 2023. paper

Zijie Li, Dule Shu, and Amir Barati Farimani.

Gaussian Process

PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations. arXiv, 2022. paper

Jiahao Zhang, Shiqi Zhang, and Guang Lin.
Solving and learning nonlinear PDEs with Gaussian processes. JCP, 2021. paper

Yifan Chen, Bamdad Hosseini, Houman Owhadi, and Andrew M.Stuart.
Neural-net-induced Gaussian process regression for function approximation and PDE solution. JCP, 2019. paper

Guofei Pang, Liu Yang, and George Em Karniadakis.
Learning neural optimal interpolation models and solvers. arXiv, 2022. paper

Maxime Beauchamp, Joseph Thompson, Hugo Georgenthum, Quentin Febvre, and Ronan Fablet.
Inference of nonlinear partial differential equations via constrained Gaussian processes. arXiv, 2022. paper

Zhaohui Li, Shihao Yang, and Jeff Wu.
Gaussian process priors for systems of linear partial differential equations with constant coefficients. arXiv, 2022. paper

Marc Härkönen, Markus Lange-Hegermann, and Bogdan Raiţă.
Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes. arXiv, 2023. paper

Yifan Chen, Houman Owhadi, and Florian Schäfer.

Solver

Solver-in-the-loop: Learning from differentiable physics to interact with iterative PDE-solvers. NIPS, 2020. paper

Kiwon Um, Robert Brand, Yun (Raymond) Fei, Philipp Holl, and Nils Thuerey.
Lie point symmetry data augmentation for neural PDE solvers. ICML, 2022. paper

Johannes Brandstetter, Max Welling, and Daniel E. Worrall.
Incorporating symmetry into deep dynamics models for improved generalization. ICLR, 2021. paper

Rui Wang, Robin Walters, and Rose Yu.
HyperSolvers: Toward fast continuous-depth models. NIPS, 2020. paper

Michael Poli, Stefano Massaroli, Atsushi Yamashita, Hajime Asama, and Jinkyoo Park.
PIXEL: Physics-informed cell representations for fast and accurate PDE solvers. NIPS, 2022. paper

Namgyu Kang, Byeonghyeon Lee, Youngjoon Hong, Seok-Bae Yun, and Eunbyung Park.
NeuralSim: Augmenting differentiable simulators with neural networks. ICRA, 2021. paper

Eric Heiden, David Millard, Erwin Coumans, Yizhou Sheng, and Gaurav S. Sukhatme.
DPM: A deep learning PDE augmentation method with application to large-eddy simulation. JCP, 2020. paper

Justin Sirignano, Jonathan F.MacArt, and Jonathan B.Freund.
General covariance data augmentation for neural PDE solvers. arXiv, 2023. paper

Vladimir Fanaskov, Tianchi Yu, Alexander Rudikov, and Ivan Oseledets.
Learning preconditioners for conjugate gradient PDE solvers. ICML, 2023. paper

Yichen Li, Peter Yichen Chen, Tao Du, and Wojciech Matusik.
Stability of implicit neural networks for long-term forecasting in dynamical systems ICLR, 2023. paper

Leon Migus, Julien Salomon, and Patrick Gallinari.

Variation

PI-VAE: Physics-informed variational auto-encoder for stochastic differential equations. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Weiheng Zhong and Hadi Meidani.
Robust SDE-based variational formulations for solving linear PDEs via deep learning. ICML, 2022. paper

Lorenz Richter and Julius Berner.
HP-VPINNs: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis.
Variational onsager neural networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs. Journal of the Mechanics and Physics of Solids, 2022. paper

Shenglin Huang, Zequn He, Bryan Chem, and Celia Reina.
Solving PDEs by variational physics-informed neural networks: A posteriori error analysis. arXiv, 2022. paper

Stefano Berrone, Claudio Canuto, and Moreno Pintore.
Variational Monte Carlo approach to partial differential equations with neural networks. arXiv, 2022. paper

Moritz Reh and Martin Gärttner.
Energetic variational neural network discretizations to gradient flows. arXiv, 2022. paper

Ziqing Hu, Chun Liu, Yiwei Wang, and Zhiliang Xu.
Variational Bayes deep operator network: A data-driven Bayesian solver for parametric differential equations. arXiv, 2022. paper

Shailesh Garg and Souvik Chakraborty.
Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems. arXiv, 2023. paper

Xuhui Meng.

Bayesian

Bayesian deep learning for partial differential equation parameter discovery with sparse and noisy data. JCP: X, 2022. paper

Christophe Bonneville and Christopher Earls.
B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. JCP, 2021. paper

Liu Yang, Xuhui Meng, and George Em Karniadakis.
Approximate Bayesian neural operators: Uncertainty quantification for parametric PDEs. arXiv, 2022. paper

Emilia Magnani, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig.
Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants. arXiv, 2022. paper

L. Mars Gao and J. Nathan Kutz.
Bayesian physics informed neural networks for data assimilation and spatio-temporal modelling of wildfires. arXiv, 2022. paper

Joel Janek Dabrowski, Daniel Edward Pagendam, James Hilton, Conrad Sanderson, Daniel MacKinlay, Carolyn Huston, Andrew Bolt, and Petra Kuhnert.

Lagrangian

Lagrangian PINNs: A causality–conforming solution to failure modes of physics-informed neural networks. arXiv, 2022. paper

Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh.
AL-PINNs: Augmented Lagrangian relaxation method for physics-informed neural networks. arXiv, 2022. paper

Hwijae Son, Sung Woong Cho, and Hyung Ju Hwang.
Lagrangian flow networks for conservation laws. arXiv, 2023. paper

Fabricio Arend Torres, Marcello Massimo Negri, Marco Inversi, and Jonathan Aellen.

Uncertainty Quantification

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. arXiv, 2021. paper

Dongkun Zhang, Lu Lu, Ling Guo, and George Em Karniadakis.
Adversarial uncertainty quantification in physics-informed neural networks. JCP, 2021. paper

Yibo Yang and Paris Perdikaris.
Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models. JCP, 2020. paper

Ramakrishna Tipireddy, David A.Barajas-Solano, and Alexandre M.Tartakovsky.
Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. JCP, 2019. paper

Yinhao Zhu, Nicholas Zabarasa, Phaedon-Stelios Koutsourelakisb, and Paris Perdikaris.
Error-aware B-PINNs: Improving uncertainty quantification in Bayesian physics-informed neural networks. arXiv, 2022. paper

Olga Graf, Pablo Flores, Pavlos Protopapas, and Karim Pichara.
Physics-informed information field theory for modeling physical systems with uncertainty quantification. arXiv, 2023. paper

Alex Alberts and Ilias Bilionis.
Quantifying uncertainty for deep learning based forecasting and flow-reconstruction using neural architecture search ensembles. arXiv, 2023. paper

Romit Maulik, Romain Egele, Krishnan Raghavan, and Prasanna Balaprakash.
Physics-informed variational inference for uncertainty quantification of stochastic differential equations. JCP, 2023. paper

Hyomin Shin and Minseok Choi.
Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons. JCP, 2023. paper

Apostolos F. Psaros, Xuhui Meng, Zongren Zou, Ling Guo, and George Em Karniadakis.

Active Learning

Neural Galerkin scheme with active learning for high-dimensional evolution equations. arXiv, 2022. paper

Joan Bruna, Benjamin Peherstorfer, and Eric Vanden-Eijnden.
Discovering and forecasting extreme events via active learning in neural operators. arXiv, 2022. paper

Ethan Pickering, Stephen Guth, George Em Karniadakis, and Themistoklis P. Sapsis.
Active learning based sampling for high-dimensional nonlinear partial differential equations. JCP, 2023. paper

Wenhan Gao and Chunmei Wang.

Multi Scale

Hierarchical deep learning of multiscale differential equation time-steppers. Philosophical Transactions of the Royal Society A, 2022. paper

Yuying Liu, J. Nathan Kutz, and Steven L. Brunton.
NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems. JCP, 2022. paper

Wing Tat Leung, Guang Lin, and Zecheng Zhang.
Deep multiscale model learning. JCP, 2020. paper

Yating Wang, Siu Wun Cheung, Eric T.Chung, Yalchin Efendiev, and Min Wang.
Multi-scale deep neural networks for solving high dimensional PDEs. arXiv, 2019. paper

Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz.
Towards multi-spatiotemporal-scale generalized PDE modeling. arXiv, 2022. paper

Jayesh K. Gupta and Johannes Brandstetter.

Multi Fidelity

Multifidelity deep operator networks. arXiv, 2022. paper

Amanda A. Howard, Mauro Perego, George Em Karniadakis, and Panos Stinis.
Physics and equality constrained artificial neural networks: Application to forward and inverse problems with multi-fidelity data fusion. JCP, 2022. paper

Lulu Zhang, Tao Luo, Yaoyu Zhang, Weinan E, Zhiqin John Xu, and Zheng Ma.
A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems. JCP, 2020. paper

Xuhui Meng and George Em Karniadakis.
Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Physical Review Research, 2022. paper

Lu Lu, Raphaël Pestourie, Steven G. Johnson, and Giuseppe Romano.

Multi Grid

Learning to optimize multigrid PDE solvers. ICML, 2019. paper

Daniel Greenfeld, Meirav Galun, Ronen Basri, Irad Yavneh, and Ron Kimmel.

Applications

Optimization

Fast PDE-constrained optimization via self-supervised operator learning. arXiv, 2021. paper

Sifan Wang, Mohamed Aziz Bhouri, and Paris Perdikaris.
An extended physics informed neural network for preliminary analysis of parametric optimal control problems. arXiv, 2021. paper

Nicola Demo, Maria Strazzullo, and Gianluigi Rozza.
Optimal control of PDEs using physics-informed neural networks. JCP, 2023. paper

Saviz Mowlavi and Saleh Nabi.
Solving PDE-constrained control problems using operator learning. AAAI, 2022. paper

Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, and Hyung Ju Hwang.
PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.
Control of partial differential equations via physics-informed neural networks. Journal of Optimization Theory and Applications, 2022. paper

Carlos J. García-Cervera, Mathieu Kessler, and Francisco Periago.
A machine learning framework for solving high-dimensional mean field game and mean field control problems. PNAS, 2020. paper

Lars Ruthottoa, Stanley J. Osherc, Wuchen Lic, Levon Nurbekyanc, and Samy Wu Fung.
Bi-level physics-informed neural networks for PDE constrained optimization using Broyden's hypergradients. ICLR, 2023. paper

Zhongkai Hao, Chengyang Ying, Hang Su, Jun Zhu, Jian Song, and Ze Cheng.
A combination technique for optimal control problems constrained by random PDEs. arXiv, 2022. paper

Fabio Nobile and Tommaso Vanzan.
A multilevel reinforcement learning framework for PDE-based control. arXiv, 2022. paper

Atish Dixit and Ahmed H. Elsheikh.
Optimal learning of high-dimensional classification problems using deep neural networks. arXiv, 2022. paper

Philipp Petersen and Felix Voigtlaender.
The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: A deep learning approach. arXiv, 2023. paper

Yongcun Song, Xiaoming Yuan, and Hangrui Yue.
Learning differentiable solvers for systems with hard constraints. ICLR, 2023. paper

Geoffrey Négiar, Geoffrey_Négiar, Michael W. Mahoney, and Aditi Krishnapriyan.
Volumetric optimal transportation by fast Fourier transform. ICLR, 2023. paper

Na Lei, DONGSHENG An, Min Zhang, Xiaoyin Xu, and David Gu.
PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.
PDE-constrained models with neural network terms: Optimization and global convergence. JCP, 2023. paper

Justin Sirignano, Jonathan MacArt, and Konstantinos Spiliopoulos.
Topology optimization using neural networks with conditioning field initialization for improved efficiency. arXiv, 2023. paper

Hongrui Chen, Aditya Joglekar, and Levent Burak Kara.
Deep reinforcement learning for optimal well control in subsurface systems with uncertain geology. JCP, 2023. paper

Yusuf Nasir and Louis J. Durlofsky.

Fluid

Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 2021. paper

Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis.
Neural operator prediction of linear instability waves in high-speed boundary layers. JCP, 2022. paper

Patricio Clark Di Leoni, Lu Lu, Charles Meneveau, George Karniadakis, and Tamer A. Zaki.
A physics-informed convolutional neural network for the simulation and prediction of two-phase darcy flows in heterogeneous porous media. JCP, 2023. paper

Zhao Zhang, Xia Yan, Piyang Liu, Kai Zhang, Renmin Han, and Sheng Wang.
DiscretizationNet: A machine-learning based solver for Navier–Stokes equations using finite volume discretization. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Rishikesh Ranade, Chris Hillb, and Jay Pathak.
Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 2020. paper

Luning Sun, Han Gao, Shaowu Pan, and Jianxun Wang.
Towards physics-informed deep learning for turbulent flow prediction. KDD, 2020. paper

Rui Wang, Karthik Kashinath, Mustafa Mustafa, Adrian Albert, and Rose Yu.
Learning to estimate and refine fluid motion with physical dynamics. ICML, 2022. paper

Mingrui Zhang, Jianhong Wang, James Tlhomole, and Matthew D. Piggott.
Physics informed neural fields for smoke reconstruction with sparse data. ACM Transactions on Graphics, 2022. paper

Mengyu Chu, Lingjie Liu, Quan Zheng, Erik Franz, Hans-Peter Seidel, Christian Theobalt, and Rhaleb Zayer.
Physics-informed deep learning for traffic state estimation: A hybrid paradigm informed by second-order traffic models. AAAI, 2021. paper

Rongye Shi, Zhaobin Mo, and Xuan Di.
Residual-based adaptivity for two-phase flow simulation in porous media using physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2022. paper

John M.Hanna, José V.Aguado, Sebastien Comas-Cardona, Ramz Askri, and Domenico Borzacchiello.
Learned turbulence modelling with differentiable fluid solvers: Physics-based loss-functions and optimisation horizons. JFM, 2022. paper

Björn List, Liwei Chen, and Nils Thuerey.
Learning hydrodynamic equations for active matter from particle simulations and experiments. PNAS, 2023. paper

Rohit Supekar, Boya Song, Alasdair Hastewell, Gary P. T. Choi, Alexander Mietke, and Jörn Dunkel.
Physics informed neural networks: A case study for gas transport problems. JCP, 2023. paper

Erik Laurin Strelow, Alf Gerisch, Jens Lang, and Marc E. Pfetsch.

Reconstruction

Occupancy networks: Learning 3D reconstruction in function space. CVPR, 2019. paper

Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger.
Transfer learning for flow reconstruction based on multifidelity data. AIAA Journal, 2022. paper

Jiaqing Kou, Chenjia Ning, and Weiwei Zhang.
Learning-based state reconstruction for a scalar hyperbolic PDE under noisy lagrangian sensing. L4DC, 2022. paper

Matthieu Barreau, John Liu, and Karl Henrik Johansson.

Physics

Dynamic weights enabled physics-informed neural network for simulating the mobility of engineered nano-particles in a contaminated aquifer. NIPS, 2022. paper

Shikhar Nilabh and Fidel Grandia.
Learning two-phase microstructure evolution using neural operators and autoencoder architectures. NPJ Computational Materials, 2022. paper

Vivek Oommen, Khemraj Shukla, Somdatta Goswami, Rémi Dingreville, and George Em Karniadakis.
Predicting glass structure by physics-informed machine learning. NPJ Computational Materials, 2022. paper

Mikkel L. Bødker, Mathieu Bauchy, Tao Du, John C. Mauro, and Morten M. Smedskjaer.
Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium. NPJ Computational Materials, 2022. paper

Ruiyang Li, Jianxun Wang, Eungkyu Lee, and Tengfei Luo.
Design of Turing systems with physics-informed neural networks. arXiv, 2022. paper

Jordon Kho, Winston Koh, Jian Cheng Wong, Pao-Hsiung Chiu, and Chin Chun Ooi.
Spatio-temporal super-resolution of dynamical systems using physics-informed deep-learning. AAAI, 2023. paper

Rajat Arora and Ankit Shrivastava.
Rapid seismic waveform modeling and inversion with neural operators. TGRS, 2023. paper

Yan Yang, Angela F. Gao, Kamyar Azizzadenesheli, Robert W. Clayton, and Zachary E. Ross.

Image

Learning to diffuse: A new perspective to design PDEs for visual analysis. TPAMI, 2016. paper

Risheng Liu, Guangyu Zhong, Junjie Cao, Zhouchen Lin, Shiguang Shan, and Zhongxuan Luo.
Reformulating optical flow to solve image-based inverse problems and quantify uncertainty. TPAMI, 2022. paper

Aleix Boquet-Pujadas and Jean-Christophe Olivo-Marin.
WarpPINN: Cine-MR image registration with physics-informed neural networks. arXiv, 2022. paper

Pablo Arratia Lopez, Hernan Mella, Sergio Uribe, Daniel E. Hurtado, and Francisco Sahli Costabal.
NODE-ImgNet: a PDE-informed effective and robust model for image denoising. arXiv, 2023. paper

Xinheng Xie, Yue Wu, Hao Nib, and Cuiyu He.

Mechanics

Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems. Computer Methods in Applied Mechanics and Engineering, 2022. paper

Tapas Tripura and Souvik Chakraborty.
Graph neural networks for airfoil design. arXiv, 2023. paper

Florent Bonnet.

Robotics

Hybrid learning of time-series inverse dynamics models for locally isotropic robot motion. RAL, 2022. paper

Tolga-Can Çallar and Sven Böttger.
NTFields: Neural time fields for physics-informed robot motion planning. ICLR, 2023. paper

Ruiqi Ni and Ahmed H Qureshi.
Online parameter estimation using physics-informed deep learning for vehicle stability algorithms. arXiv, 2023. paper

Kemal Koysuren, Ahmet Faruk Keles, and Melih Cakmakci.

Cybernetics

Machine learning accelerated PDE backstepping observers. arXiv, 2022. paper

Yuanyuan Shi, Zongyi Li, Huan Yu, Drew Steeves, Anima Anandkumar, and Miroslav Krstic.
Neural solvers for fast and accurate numerical optimal control. NIPS, 2021. paper

Federico Berto, Stefano Massaroli, Michael Poli, and Jinkyoo Park.
Bellman neural networks for the class of optimal control problems with integral quadratic cost. TAI, 2022. paper

Enrico Schiassi, Andrea D'Ambrosio, and Roberto Furfaro.
Offline supervised learning vs online direct policy optimization: A comparative study and a unifie training paradigm for neural network-based optimal feedback control. arXiv, 2022. paper

Yue Zhao and Jiequn Han.
Policy evaluation and temporal–difference learning in continuous time and space: A martingale approach. JMLR, 2022. paper

Yanwei Jia and Xunyu Zhou.
Physics-informed kernel embeddings: Integrating prior system knowledge with data-driven control. arXiv, 2023. paper

Adam J. Thorpe, Cyrus Neary, Franck Djeumou, Meeko M. K. Oishi, and Ufuk Topcu.
Distributed control of partial differential equations using convolutional reinforcement learning. arXiv, 2023. paper

Sebastian Peitz, Jan Stenner, Vikas Chidananda, Oliver Wallscheid, Steven L. Brunton, and Kunihiko Taira.
Neural control of parametric solutions for high-dimensional evolution PDEs. arXiv, 2023. paper

Nathan Gaby, Xiaojing Ye, and Haomin Zhou.
Bridging physics-informed neural networks with reinforcement learning: Hamilton-Jacobi-Bellman proximal policy optimization (HJBPPO). arXiv, 2023. paper

Amartya Mukherjee and Jun Liu.
AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems. arXiv, 2023. paper

Pengfei Yin, Guangqiang Xiao, Kejun Tang, and Chao Yang.
Neural operators for bypassing gain and control computations in PDE backstepping. arXiv, 2023. paper

Luke Bhan, Yuanyuan Shi, and Miroslav Krstic.
Neural operators of backstepping controller and observer gain functions for reaction-diffusion PDEs. arXiv, 2023. paper

Miroslav Krstic, Luke Bhan, and Yuanyuan Shi.
Leveraging multi-time Hamilton-Jacobi PDEs for certain scientific machine learning problems. arXiv, 2023. paper

Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, and George Em Karniadakis.
Learning to control PDEs with differentiable physics. ICLR, 2020. paper

Philipp Holl, Nils Thuerey, and Vladlen Koltun.
A generalizable physics-informed learning framework for risk probability estimation. L4DC, 2020. paper

Zhuoyuan Wang and Yorie Nakahira.
Operator learning for nonlinear adaptive control. L4DC, 2023. paper

Luke Bhan, Yuanyuan Shi and Miroslav Krstic.

Inverse Design

DPM: Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems. AAAI, 2021. paper

Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jin, and Noseong Park.
Neural inverse operators for Solving PDE inverse problems. arXiv, 2023. paper

Roberto Molinaro, Yunan Yang, Björn Engquist, and Siddhartha Mishra.
Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems. JCP, 2022. paper

Jing Li and Alexandre M.Tartakovsky.
Solving inverse problems in stochastic models using deep neural networks and adversarial training. Computer Methods in Applied Mechanics and Engineering, 2021. paper

Kailai Xu and Eric Darve.
Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 2021. paper

Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G. Johnson.
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. JCP, 2019. paper

M.Raissia, P.Perdikarisb, and George Em Karniadakis.
GRIDS-Net: Inverse shape design and identification of scatterers via geometric regularization and physics-embedded deep learning. arXiv, 2023. paper

Siddharth Nair, Timothy F. Walsh, Greg Pickrell, and Fabio Semperlotti.
Bayesian inversion with neural operator (BINO) for modeling subdiffusion: Forward and inverse problems. arXiv, 2022. paper

Xiongbin Yan, Zhiqin John Xu, and Zheng Ma.
Maximum-likelihood estimators in physics-informed neural networks for high-dimensional inverse problems. arXiv, 2023. paper

Gabriel S. Gusmão and Andrew J. Medford.

Quantum

Physics-informed neural networks for quantum eigenvalue problems. IJCNN, 2022. paper

Henry Jin, Marios Mattheakis, and Pavlos Protopapas.
Quantum-inspired tensor neural networks for partial differential equations. arXiv, 2022. paper

Raj Patel, Chia-Wei Hsing, Serkan Sahin, Saeed S. Jahromi, Samuel Palmer, Shivam Sharma, Christophe Michel, Vincent Porte, Mustafa Abid, Stephane Aubert, Pierre Castellani, Chi-Guhn Lee, Samuel Mugel, and Roman Orus.

Climate

FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators. arXiv, 2022. paper

Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar.
Fourier neural operators for arbitrary resolution climate data downscaling. JMLR, 2023. paper

Qidong Yang, Alex Hernandez-Garcia, Paula Harder, Venkatesh Ramesh, Prasanna Sattegeri, Daniela Szwarcman, Campbell D. Watson, and David Rolnick.
Modelling atmospheric dynamics with spherical Fourier neural operators. ICLR, 2023. paper

Boris Bonev, Thorsten Kurth, Christian Hundt, Jaideep Pathak, Maximilian Baust, Karthik Kashinath, and Anima Anandkumar.
Spatiotemporal modeling of European paleoclimate using doubly sparse Gaussian processes. NIPS, 2022. paper

Seth D. Axen, Alexandra Gessner, Christian Sommer, Nils Weitzel, and Álvaro Tejero-Cantero.

Game Theory

Approximating discontinuous Nash equilibria values of two-player general-sum differential games. arXiv, 2022. paper

Lei Zhang, Mukesh Ghimire, Wenlong Zhang, Zhe Xu, and Yi Ren.

Manufacturing

Physics-aware machine learning surrogates for real-time manufacturing digital twin. Manufacturing Letters, 2022. paper

Aditya Balu, Soumik Sarkar, Baskar Ganapathysubramanian, and Adarsh Krishnamurthy.
Multi-scale digital twin: Developing a fast and physics-informed surrogate model for groundwater contamination with uncertain climate models. arXiv, 2022. paper

Lijing Wang, Takuya Kurihana, Aurelien Meray, Ilijana Mastilovic, Satyarth Praveen, Zexuan Xu, Milad Memarzadeh, Alexander Lavin, and Haruko Wainwright.
SciAI4Industry--Solving PDEs for industry-scale problems with deep learning. arXiv, 2022. paper

Philipp A. Witte, Russell J. Hewett, Kumar Saurabh, AmirHossein Sojoodi, and Ranveer Chandra.
Operator learning framework for digital twin and complex engineering systems. arXiv, 2023. paper

Kazuma Kobayashi, James Daniell, and Syed B. Alam.

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