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Neural-PDE-Solver

PDE: Partial Differentiable Equation

Contributed by Chunyang Zhang.

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
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    Amuthan A. Ramabathiran and Prabhu Ramachandran.

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    Jaroslaw Rzepecki, Daniel Bates, and Chris Doran.

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  33. Momentum diminishes the effect of spectral bias in physics-informed neural networks. arXiv, 2022. paper

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  34. Δ-PINNs: Physics-informed neural networks on complex geometries. arXiv, 2022. paper

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    Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, and Kenji Kawaguchi.

  38. Physics-informed neural networks for operator equations with stochastic data. arXiv, 2022. paper

    Paul Escapil-Inchauspé and Gonzalo A. Ruz.

  39. Physics-informed neural networks with unknown measurement noise. arXiv, 2022. paper

    Philipp Pilar and Niklas Wahlstrom.

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

    Kuangdai Leng and Jeyan Thiyagalingam.

  41. 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.

  42. L-HYDRA: Multi-head physics-informed neural networks. arXiv, 2023. paper

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  1. 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.

  2. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. SA, 2021. paper

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    Sifan Wang and Paris Perdikaris.

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    Sifan Wang, Hanwen Wang, and Paris Perdikaris.

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    Wuzhe Xu, Yulong Lu, and Li Wang.

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    Nikola Kovachki, Samuel Lanthaler, and Siddhartha Mishra.

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    Francesco Alesiani, Makoto Takamoto, and Mathias Niepert.

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    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.

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    Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar.

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    Sifan Wang, Hanwen Wang, and Paris Perdikaris.

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    Michael Rotman, Amit Dekel, Ran Ilan Ber, Lior Wolf, and Yaron Oz.

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  14. Incremental spectral learning Fourier neural operator. arXiv, 2022. paper

    Jiawei Zhao, Robert Joseph George, Yifei Zhang, Zongyi Li, and Anima Anandkumar.

  15. 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.

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    Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z, and Li Cari.

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    Bilal Thonnam Thodi, Sai Venkata Ramana Ambadipudi, and Saif Eddin Jabari.

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    Ning Liu, Siavash Jafarzadeh, and Yue Yu.

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  5. 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.

  6. Learning the solution operator of boundary value problems using graph neural networks. ICML, 2022. paper

    Winfried Lötzsch, Simon Ohler, and Johannes S. Otterbach.

  7. 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.

  8. Modular flows: Differential molecular generation. NIPS, 2022. paper

    Yogesh Verma, Samuel Kaski, Markus Heinonen, and Vikas Garg.

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    Zhao Qingqing, David B. Lindell, and Gordon Wetzstein.

  10. Physics-embedded neural networks: E(n)-equivariant graph neural PDE solvers. NIPS, 2022. paper

    Masanobu Horie and Naoto Mitsume.

  11. PDE-GCN: Novel architectures for graph neural networks motivated by partial differential equations. ICLR, 2021. paper

    Moshe Eliasof, Eldad Haber, and Eran Treister.

  12. Physics-aware difference graph networks for sparsely-observed dynamics. ICLR, 2020. paper

    Sungyong Seo, Chuizheng Meng, and Yan Liu.

  13. 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.

  14. Learning continuous-time PDEs from sparse data with graph neural networks. ICLR, 2021. paper

    Valerii Iakovlev, Markus Heinonen, and Harri Lähdesmäki.

  15. 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.

  16. 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.

  17. 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.

  18. GRAND: Graph neural diffusion. ICML, 2021. paper

    Benjamin Paul Chamberlain, James Rowbottom, Maria I. Gorinova, Stefan D Webb, Emanuele Rossi, and Michael M. Bronstein.

  19. 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.

  20. Neural networks trained to solve differential equations learn general representations. NIPS, 2018. paper

    Martin Magill, Faisal Qureshi, and Hendrick de Haan.

  21. 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.

  22. Physics-constrained unsupervised learning of partial differential equations using meshes. arXiv, 2022. paper

    Mike Y. Michelis and Robert K. Katzschmann.

  23. 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.

  24. Bi-stride multi-scale graph neural network for mesh-based physical simulation. arXiv, 2022. paper

    Yadi Cao, Menglei Chai, Minchen Li, and Chenfanfu Jiang.

  25. Learning time-dependent PDE solver using message passing graph neural networks. arXiv, 2022. paper

    Pourya Pilva and Ahmad Zareei.

  26. 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.

  27. STONet: A neural-operator-driven spatio-temporal network. arXiv, 2022. paper

    Haitao Lin, Guojiang Zhao, Lirong Wu, and Stan Z. Li.

  28. PhyGNNet: Solving spatiotemporal PDEs with physics-informed graph neural network. arXiv, 2022. paper

    Longxiang Jiang, Liyuan Wang, Xinkun Chu, Yonghao Xiao, and Hao Zhang.

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    Sungyong Seo and Yan Liu.

  30. 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.

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    Léonard Equer, T. Konstantin Rusch, and Siddhartha Mishra.

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  33. GNN-based physics solver for time-independent PDEs. arXiv, 2023. paper

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

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

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    Pratyush Bhatt, Yash Kumar, and Azzeddine Soulaimani.

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

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  2. Variational autoencoding neural operators. arXiv, 2023. paper

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

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  2. On the representation of solutions to elliptic PDEs in Barron space. NIPS, 2021. paper

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  3. Low-rank registration based manifolds for convection-dominated PDEs. AAAI, 2021. paper

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  7. On neurosymbolic solutions for PDEs. arXiv, 2022. paper

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  11. NOMAD: Nonlinear manifold decoders for operator learning. arXiv, 2022. paper

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  12. U-NO: U-shaped neural operators. arXiv, 2022. paper

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  13. Wavelet neural operator: A neural operator for parametric partial differential equations. arXiv, 2022. paper

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  14. Pseudo-differential integral operator for learning solution operators of partial differential equations. arXiv, 2022. paper

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  15. Nonlinear reconstruction for operator learning of PDEs with discontinuities. arXiv, 2022. paper

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  16. GeONet: A neural operator for learning the Wasserstein geodesic. arXiv, 2022. paper

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  17. Render unto numerics: Orthogonal polynomial neural operator for PDEs with non-periodic boundary conditions. arXiv, 2022. paper

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  18. 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.

  19. Guiding continuous operator learning through physics-based boundary constraints. arXiv, 2022. paper

    Nadim Saad, Gaurav Gupta, Shima Alizadeh, and Danielle C. Maddix.

  20. BelNet: Basis enhanced learning, a mesh-free neural operator. arXiv, 2022. paper

    Zecheng Zhang, Wing Tat Leung, and Hayden Schaeffer.

  21. INO: Invariant neural operators for learning complex physical systems with momentum conservation. arXiv, 2022. paper

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  22. BINN: A deep learning approach for computational mechanics problems based on boundary integral equations. arXiv, 2023. paper

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  23. Koopman neural operator as a mesh-free solver of non-linear partial differential equations. arXiv, 2023. paper

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  25. Deep operator learning lessens the curse of dimensionality for PDEs. arXiv, 2023. paper

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  22. Temporal consistency loss for physics-informed neural networks. arXiv, 2023. paper

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  23. Can physics-informed neural networks beat the finite element method? arXiv, 2023. paper

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  24. 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.

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  2. CROM: Continuous reduced-order modeling of PDEs using implicit neural representations. ICLR, 2023. paper

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  16. Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios. arXiv, 2022. paper

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  18. Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems. arXiv, 2023. paper

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  4. 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.

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

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  2. DMIS: Dynamic mesh-based importance sampling for training physics-informed neural networks. AAAI, 2023. paper

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  3. 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

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    Arka Daw, Jie Bu, Sifan Wang, Paris Perdikaris, and Anuj Karpatne.

  5. Residual-quantile adjustment for adaptive training of physics-informed neural network. arXiv, 2022. paper

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

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  2. A latent space solver for PDE generalization. ICLR, 2021. paper

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  3. Approximate latent force model inference. AAAI, 2021. paper

    Jacob D. Moss, Felix L. Opolka, Bianca Dumitrascu, and Pietro Lió.

  4. Learning to accelerate partial differential equations via latent global evolution. NIPS, 2022. paper

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  5. Exploring physical latent spaces for deep learning. arXiv, 2022. paper

    Chloe Paliard, Nils Thuerey, and Kiwon Um.

  6. 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.

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  8. Learning in latent spaces improves the predictive accuracy of deep neural operators. arXiv, 2023. paper

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

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  2. Adaptive deep neural networks methods for high-dimensional partial differential equations. JCP, 2022. paper

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  3. Self-adaptive loss balanced Physics-informed neural networks. Neurocomputing, 2022. paper

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  4. Multi-objective loss balancing for physics-informed deep learning. arXiv, 2021. paper

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  5. 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.

  6. 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.

  7. Loss landscape engineering via data regulation on PINNs. arXiv, 2022. paper

    Vignesh Gopakumar, Stanislas Pamela, and Debasmita Samaddar.

  8. Implicit stochastic gradient descent for training physics-informed neural networks. AAAI, 2023. paper

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  9. 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.

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

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  2. Learning composable energy surrogates for PDE order reduction. NIPS, 2020. paper

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  3. Neural basis functions for accelerating solutions to high mach euler equations. ICML, 2022. paper

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  4. PFNN-2: A domain decomposed penalty-free neural network method for solving partial differential equations. arXiv, 2022. paper

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

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  6. Finite basis physics-informed neural networks as a Schwarz domain decomposition method. arXiv, 2022. paper

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  7. NeuralStagger: Accelerating physics constrained neural PDE solver with spatialtemporal decomposition. arXiv, 2023. paper

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  11. Physics-informed spectral learning: the Discrete Helmholtz-Hodge decomposition. arXiv, 2023. paper

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

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  2. 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.

  3. RANG: A residual-based adaptive node generation method for physics-informed neural networks. arXiv, 2022. paper

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  4. Learning a mesh motion technique with application to fluid-structure interaction and shape optimization. arXiv, 2022. paper

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  5. Accelerated training of physics-informed neural networks (PINNs) using meshless discretizations. arXiv, 2022. paper

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  6. An improved structured mesh generation method based on physics-informed neural networks. arXiv, 2022. paper

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

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    Gavin Kerrigan, Justin Ley, and Padhraic Smyth.

  10. 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.

  11. PIAT: Physics informed adversarial training for solving partial differential equations. arXiv, 2022. paper

    Simin Shekarpaz, Mohammad Azizmalayeri, and Mohammad Hossein Rohban.

  12. Physics-constrained generative adversarial networks for 3D turbulence. arXiv, 2022. paper

    Dima Tretiak, Arvind T. Mohan, and Daniel Livescu.

  13. 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.

  14. Infinite-dimensional diffusion models for function spaces. arXiv, 2023. paper

    Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang.

  15. A physics-informed diffusion model for high-fidelity flow field reconstruction. JCP, 2023. paper

    Dule Shu, Zijie Li, and Amir Barati Farimani.

  16. Generative modelling with inverse heat dissipation. ICLR, 2023. paper

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  19. ViTO: Vision Transformer-operator. arXiv, 2023. paper

    Oded Ovadia, Adar Kahana, Panos Stinis, Eli Turkel, and George Em Karniadakis.

  20. LatentPINNs: Generative physics-informed neural networks via a latent representation learning. arXiv, 2023. paper

    Mohammad H. Taufik and Tariq Alkhalifah.

  21. Generative diffusion learning for parametric partial differential equations. arXiv, 2023. paper

    Ting Wang, Petr Plechac, and Jaroslaw Knap.

  22. Scalable Transformer for PDE surrogate modeling. arXiv, 2023. paper

    Zijie Li, Dule Shu, and Amir Barati Farimani.

  1. 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.

  2. Solving and learning nonlinear PDEs with Gaussian processes. JCP, 2021. paper

    Yifan Chen, Bamdad Hosseini, Houman Owhadi, and Andrew M.Stuart.

  3. Neural-net-induced Gaussian process regression for function approximation and PDE solution. JCP, 2019. paper

    Guofei Pang, Liu Yang, and George Em Karniadakis.

  4. Learning neural optimal interpolation models and solvers. arXiv, 2022. paper

    Maxime Beauchamp, Joseph Thompson, Hugo Georgenthum, Quentin Febvre, and Ronan Fablet.

  5. Inference of nonlinear partial differential equations via constrained Gaussian processes. arXiv, 2022. paper

    Zhaohui Li, Shihao Yang, and Jeff Wu.

  6. Gaussian process priors for systems of linear partial differential equations with constant coefficients. arXiv, 2022. paper

    Marc Härkönen, Markus Lange-Hegermann, and Bogdan Raiţă.

  7. Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes. arXiv, 2023. paper

    Yifan Chen, Houman Owhadi, and Florian Schäfer.

  1. 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.

  2. Lie point symmetry data augmentation for neural PDE solvers. ICML, 2022. paper

    Johannes Brandstetter, Max Welling, and Daniel E. Worrall.

  3. Incorporating symmetry into deep dynamics models for improved generalization. ICLR, 2021. paper

    Rui Wang, Robin Walters, and Rose Yu.

  4. HyperSolvers: Toward fast continuous-depth models. NIPS, 2020. paper

    Michael Poli, Stefano Massaroli, Atsushi Yamashita, Hajime Asama, and Jinkyoo Park.

  5. 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.

  6. NeuralSim: Augmenting differentiable simulators with neural networks. ICRA, 2021. paper

    Eric Heiden, David Millard, Erwin Coumans, Yizhou Sheng, and Gaurav S. Sukhatme.

  7. 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.

  8. General covariance data augmentation for neural PDE solvers. arXiv, 2023. paper

    Vladimir Fanaskov, Tianchi Yu, Alexander Rudikov, and Ivan Oseledets.

  9. Learning preconditioners for conjugate gradient PDE solvers. ICML, 2023. paper

    Yichen Li, Peter Yichen Chen, Tao Du, and Wojciech Matusik.

  10. Stability of implicit neural networks for long-term forecasting in dynamical systems ICLR, 2023. paper

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  1. 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.

  2. Robust SDE-based variational formulations for solving linear PDEs via deep learning. ICML, 2022. paper

    Lorenz Richter and Julius Berner.

  3. 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.

  4. 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.

  5. Solving PDEs by variational physics-informed neural networks: A posteriori error analysis. arXiv, 2022. paper

    Stefano Berrone, Claudio Canuto, and Moreno Pintore.

  6. Variational Monte Carlo approach to partial differential equations with neural networks. arXiv, 2022. paper

    Moritz Reh and Martin Gärttner.

  7. Energetic variational neural network discretizations to gradient flows. arXiv, 2022. paper

    Ziqing Hu, Chun Liu, Yiwei Wang, and Zhiliang Xu.

  8. Variational Bayes deep operator network: A data-driven Bayesian solver for parametric differential equations. arXiv, 2022. paper

    Shailesh Garg and Souvik Chakraborty.

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

    Xuhui Meng.

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

    Christophe Bonneville and Christopher Earls.

  2. 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.

  3. Approximate Bayesian neural operators: Uncertainty quantification for parametric PDEs. arXiv, 2022. paper

    Emilia Magnani, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig.

  4. Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants. arXiv, 2022. paper

    L. Mars Gao and J. Nathan Kutz.

  5. 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.

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

    Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh.

  2. AL-PINNs: Augmented Lagrangian relaxation method for physics-informed neural networks. arXiv, 2022. paper

    Hwijae Son, Sung Woong Cho, and Hyung Ju Hwang.

  3. Lagrangian flow networks for conservation laws. arXiv, 2023. paper

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

  1. 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.

  2. Adversarial uncertainty quantification in physics-informed neural networks. JCP, 2021. paper

    Yibo Yang and Paris Perdikaris.

  3. 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.

  4. 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.

  5. 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.

  6. Physics-informed information field theory for modeling physical systems with uncertainty quantification. arXiv, 2023. paper

    Alex Alberts and Ilias Bilionis.

  7. 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.

  8. Physics-informed variational inference for uncertainty quantification of stochastic differential equations. JCP, 2023. paper

    Hyomin Shin and Minseok Choi.

  9. 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.

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

    Joan Bruna, Benjamin Peherstorfer, and Eric Vanden-Eijnden.

  2. 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.

  3. Active learning based sampling for high-dimensional nonlinear partial differential equations. JCP, 2023. paper

    Wenhan Gao and Chunmei Wang.

  1. 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.

  2. NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems. JCP, 2022. paper

    Wing Tat Leung, Guang Lin, and Zecheng Zhang.

  3. Deep multiscale model learning. JCP, 2020. paper

    Yating Wang, Siu Wun Cheung, Eric T.Chung, Yalchin Efendiev, and Min Wang.

  4. 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.

  5. Towards multi-spatiotemporal-scale generalized PDE modeling. arXiv, 2022. paper

    Jayesh K. Gupta and Johannes Brandstetter.

  1. Multifidelity deep operator networks. arXiv, 2022. paper

    Amanda A. Howard, Mauro Perego, George Em Karniadakis, and Panos Stinis.

  2. 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.

  3. 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.

  4. 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.

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

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

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

    Sifan Wang, Mohamed Aziz Bhouri, and Paris Perdikaris.

  2. An extended physics informed neural network for preliminary analysis of parametric optimal control problems. arXiv, 2021. paper

    Nicola Demo, Maria Strazzullo, and Gianluigi Rozza.

  3. Optimal control of PDEs using physics-informed neural networks. JCP, 2023. paper

    Saviz Mowlavi and Saleh Nabi.

  4. Solving PDE-constrained control problems using operator learning. AAAI, 2022. paper

    Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, and Hyung Ju Hwang.

  5. PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

    Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.

  6. 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.

  7. 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.

  8. 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.

  9. A combination technique for optimal control problems constrained by random PDEs. arXiv, 2022. paper

    Fabio Nobile and Tommaso Vanzan.

  10. A multilevel reinforcement learning framework for PDE-based control. arXiv, 2022. paper

    Atish Dixit and Ahmed H. Elsheikh.

  11. Optimal learning of high-dimensional classification problems using deep neural networks. arXiv, 2022. paper

    Philipp Petersen and Felix Voigtlaender.

  12. The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: A deep learning approach. arXiv, 2023. paper

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue.

  13. Learning differentiable solvers for systems with hard constraints. ICLR, 2023. paper

    Geoffrey Négiar, Geoffrey_Négiar, Michael W. Mahoney, and Aditi Krishnapriyan.

  14. Volumetric optimal transportation by fast Fourier transform. ICLR, 2023. paper

    Na Lei, DONGSHENG An, Min Zhang, Xiaoyin Xu, and David Gu.

  15. PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

    Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.

  16. PDE-constrained models with neural network terms: Optimization and global convergence. JCP, 2023. paper

    Justin Sirignano, Jonathan MacArt, and Konstantinos Spiliopoulos.

  17. Topology optimization using neural networks with conditioning field initialization for improved efficiency. arXiv, 2023. paper

    Hongrui Chen, Aditya Joglekar, and Levent Burak Kara.

  18. Deep reinforcement learning for optimal well control in subsurface systems with uncertain geology. JCP, 2023. paper

    Yusuf Nasir and Louis J. Durlofsky.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. Towards physics-informed deep learning for turbulent flow prediction. KDD, 2020. paper

    Rui Wang, Karthik Kashinath, Mustafa Mustafa, Adrian Albert, and Rose Yu.

  7. Learning to estimate and refine fluid motion with physical dynamics. ICML, 2022. paper

    Mingrui Zhang, Jianhong Wang, James Tlhomole, and Matthew D. Piggott.

  8. 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.

  9. 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.

  10. 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.

  11. Learned turbulence modelling with differentiable fluid solvers: Physics-based loss-functions and optimisation horizons. JFM, 2022. paper

    Björn List, Liwei Chen, and Nils Thuerey.

  12. 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.

  13. 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.

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

    Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger.

  2. Transfer learning for flow reconstruction based on multifidelity data. AIAA Journal, 2022. paper

    Jiaqing Kou, Chenjia Ning, and Weiwei Zhang.

  3. Learning-based state reconstruction for a scalar hyperbolic PDE under noisy lagrangian sensing. L4DC, 2022. paper

    Matthieu Barreau, John Liu, and Karl Henrik Johansson.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. Spatio-temporal super-resolution of dynamical systems using physics-informed deep-learning. AAAI, 2023. paper

    Rajat Arora and Ankit Shrivastava.

  7. 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.

  1. 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.

  2. Reformulating optical flow to solve image-based inverse problems and quantify uncertainty. TPAMI, 2022. paper

    Aleix Boquet-Pujadas and Jean-Christophe Olivo-Marin.

  3. 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.

  4. NODE-ImgNet: a PDE-informed effective and robust model for image denoising. arXiv, 2023. paper

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

  1. 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.

  2. Graph neural networks for airfoil design. arXiv, 2023. paper

    Florent Bonnet.

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

    Tolga-Can Çallar and Sven Böttger.

  2. NTFields: Neural time fields for physics-informed robot motion planning. ICLR, 2023. paper

    Ruiqi Ni and Ahmed H Qureshi.

  3. Online parameter estimation using physics-informed deep learning for vehicle stability algorithms. arXiv, 2023. paper

    Kemal Koysuren, Ahmet Faruk Keles, and Melih Cakmakci.

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

    Yuanyuan Shi, Zongyi Li, Huan Yu, Drew Steeves, Anima Anandkumar, and Miroslav Krstic.

  2. Neural solvers for fast and accurate numerical optimal control. NIPS, 2021. paper

    Federico Berto, Stefano Massaroli, Michael Poli, and Jinkyoo Park.

  3. 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.

  4. 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.

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

    Yanwei Jia and Xunyu Zhou.

  6. 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.

  7. 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.

  8. Neural control of parametric solutions for high-dimensional evolution PDEs. arXiv, 2023. paper

    Nathan Gaby, Xiaojing Ye, and Haomin Zhou.

  9. Bridging physics-informed neural networks with reinforcement learning: Hamilton-Jacobi-Bellman proximal policy optimization (HJBPPO). arXiv, 2023. paper

    Amartya Mukherjee and Jun Liu.

  10. 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.

  11. Neural operators for bypassing gain and control computations in PDE backstepping. arXiv, 2023. paper

    Luke Bhan, Yuanyuan Shi, and Miroslav Krstic.

  12. Neural operators of backstepping controller and observer gain functions for reaction-diffusion PDEs. arXiv, 2023. paper

    Miroslav Krstic, Luke Bhan, and Yuanyuan Shi.

  13. 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.

  14. Learning to control PDEs with differentiable physics. ICLR, 2020. paper

    Philipp Holl, Nils Thuerey, and Vladlen Koltun.

  15. A generalizable physics-informed learning framework for risk probability estimation. L4DC, 2020. paper

    Zhuoyuan Wang and Yorie Nakahira.

  16. Operator learning for nonlinear adaptive control. L4DC, 2023. paper

    Luke Bhan, Yuanyuan Shi and Miroslav Krstic.

  1. 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.

  2. Neural inverse operators for Solving PDE inverse problems. arXiv, 2023. paper

    Roberto Molinaro, Yunan Yang, Björn Engquist, and Siddhartha Mishra.

  3. Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems. JCP, 2022. paper

    Jing Li and Alexandre M.Tartakovsky.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. Bayesian inversion with neural operator (BINO) for modeling subdiffusion: Forward and inverse problems. arXiv, 2022. paper

    Xiongbin Yan, Zhiqin John Xu, and Zheng Ma.

  9. Maximum-likelihood estimators in physics-informed neural networks for high-dimensional inverse problems. arXiv, 2023. paper

    Gabriel S. Gusmão and Andrew J. Medford.

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

    Henry Jin, Marios Mattheakis, and Pavlos Protopapas.

  2. 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.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  1. 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.

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

    Aditya Balu, Soumik Sarkar, Baskar Ganapathysubramanian, and Adarsh Krishnamurthy.

  2. 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.

  3. 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.

  4. Operator learning framework for digital twin and complex engineering systems. arXiv, 2023. paper

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