Physics-Informed Neural Networks (PINN) for Solving Differential Equations
Supervised by Professor Edie Miglio, Politecnico di Milano - Numerical Analysis for Machine Learning
Final Grade: 30 Cum Laude / 30
Replicating and Implementing the results of the paper: “DeepXDE: A Deep Learning Library for Solving Differential Equations” by Lu Lu, Xuhui Meng, Zhiping Mao and George Em Karniadakis.
Everything is implemented, as much as possible, from scratch, using Numpy, Scipy and JAX (no PyTorch, TensorFlow or Keras). A second version with DeepXDE library is also provided.
Key Features:
- Utilization of Physics-Informed Neural Networks (PINNs) for solving differential equations.
- Implementation of specific techniques such as hard constraints, BFGS method for optimzation and Residual Adaptive Refinement (RAR) sampling.
- Addressing diverse types of differential equations including Volterra equation (Integral Differential Equation), Kirchhoff's equation for RLC circuits (Inverse Problem), and Burgers' equation for viscous fluids.
This project explores the capabilities of Physics-Informed Neural Networks (PINN) in solving a variety of differential equations, particularly Partial Differential Equations (PDEs). The project is divided into three main parts, each focusing on a different type of differential equation and inverse problems, utilizing specific techniques to achieve accurate and efficient solutions.
Part 1: Volterra Equation and Implicit Differential Equation (IDE) In the initial phase of the project, we tackle the Volterra equation and an Integro-Differential Equation (IDE) using PINNs. Employing hard constraints and a feedforward neural network architecture, we demonstrate the effectiveness of PINNs in solving these types of equations with high precision.
Part 2: Kirchhoff's Equation for RLC Circuits (Inverse Problem) Moving forward, we delve into the realm of inverse problems by addressing Kirchhoff's equation for RLC circuits. In this phase, the PINN learns both the solution to the equation and the capacitance of the capacitor simultaneously. Leveraging a Residual Network (ResNet), we showcase how PINNs can effectively handle inverse problems, offering solutions with high precision and robustness.
Part 3: Burgers' Equation for Viscous Fluids The final segment of the project focuses on Burgers' equation, a complex, nonlinear partial differential equation governing viscous fluid flow in a tube. Here, we employ the Residual Adaptive Refinement (RAR) sampling technique to capture the impulses within the central region of the fluid tube. By tackling this challenging problem, we demonstrate the versatility and adaptability of PINNs in handling complex fluid dynamics scenarios.
In conclusion, through this project, we showcase the power of PINNs in looking at a wide range of differential equations encountered in various scientific and engineering domains. By combining neural networks with physical principles, we pave the way for innovative solutions to complex problems, offering new insights and possibilities in the field of computational physics and engineering.