Physics-Informed Neural Networks (PINNs)

Project Goal

The goal of this project is to develop a framework using a deep learning model to approximate solutions to Partial Differential Equations (PDEs) without requiring any input data. The Physics-Informed Neural Network (PINN) is trained solely on:

• Physics Loss: Derived from the governing equations (e.g., PDE = 0).
• Boundary Conditions (BC) Loss: Ensuring the solution satisfies the boundary constraints.

Objectives

The project progresses through increasingly complex problems:

1. Simple ODEs ✅:

   • PINN: $f_{\theta}: \mathbb{R} \rightarrow \mathbb{R}$
   • ODE to approximate: $f' = f, f(0) = 1$
   • Physics Loss: $\lVert f_{\theta}' - f_{\theta}\rVert$
   • Boundary Loss: $\lVert f_{\theta}(0) - 1 \rVert$
   • Analytical solution: $\exp: \mathbb{R} \rightarrow \mathbb{R}$

2. Higher-Order ODEs ✅:

   • PINN: $f_{\theta}: \mathbb{R} \rightarrow \mathbb{R}$
   • ODE to approximate: $f'' = f, f(0) = 1, f'(0) = 0$
   • Physics Loss: $\lVert f_{\theta}'' - f_{\theta} \rVert$
   • Boundary Loss: $\lVert f_{\theta}(0) - 1 \rVert, \lVert f'_{\theta}(0) \rVert$
   • Analytical solution: $\cos: \mathbb{R} \rightarrow \mathbb{R}$

3. Laplace Equation ✅:

   • PINN: $f_{\theta}: [0, 1]^2 \rightarrow \mathbb{R}$
   • PDE to approximate: $\Delta f = 0$
   • Dirichlet boundary conditions: $f(\cdot, 0) = 0, f(\cdot, 1) = \sin(\pi x), f(0, \cdot) = 0, f(1, \cdot) = 0$
   • Physics loss: $\lVert \Delta f_{\theta} \rVert$
   • Boundary loss: $\lVert f_{\theta}(\cdot, 0) \rVert, \lVert f_{\theta}(\cdot, 1) - \sin(\pi x) \rVert, \lVert f_{\theta}(0, \cdot) \rVert, \lVert f_{\theta}(1, \cdot) \rVert$
   • Analytical solution: $f(x, y) = \sin(\pi x) \sinh(\pi y)/\sinh(\pi)$

4. Euler Equations (potential, irrotational flow) ✅:

   • Wind tunnel scenario with no geometry
   • PINN: $\phi_{\theta}: [0, 4] \times [0, 1] \rightarrow \mathbb{R}$
   • PDE to approximate: None
   • Dirichlet boundary conditions: $(u, v)(\partial([0, 4] \times [0, 1])) = (1, 0)$

5. Navier-Stokes Equations ❌:

   • Solve fluid dynamics problems governed by the Navier-Stokes equations.

Installation

Ensure you have the following installed on your system:

• ✔️ Python 3.12
• ✔️ MPS support for macOS
• ✔️ Additional Python dependencies

1. Clone the Repository:

   Clone this repository to your local machine:

   git clone https://github.com/LeonDeligny/LearnPDEs.git
   cd LearnPDEs
   

2. Install dependencies:

   Install the dependencies with:

   conda env create -f environment.yml