Solving Partial Differential Equations with Physics-Informed Neural Networks (PINNs): An Introduction to Keras Implementation

This article introduces the open-source project PINNs-Keras-PDE-Solver maintained by arshadafzal, which implements PINNs technology based on Keras and can solve classic PDE problems such as the Burgers equation and Poisson equation, suitable for academic research and teaching. Project link: https://github.com/arshadafzal/PINNs-Keras-PDE-Solver, released on May 29, 2026.

Partial Differential Equations (PDEs) are the mathematical language for describing natural phenomena, but traditional numerical methods (such as finite difference and finite element methods) require complex mesh generation and large computational resources. The recently emerging Physics-Informed Neural Networks (PINNs) embed physical laws into the loss function and can solve PDEs without meshes, providing new ideas for complex physical problems.

PINNs were proposed by Raissi et al. in 2019. The core idea is to train a neural network to approximate the solution of a PDE while satisfying three constraints: control equation constraints, initial condition constraints, and boundary condition constraints. Compared to traditional methods, PINNs have advantages such as mesh-free, continuous solutions, inverse problem friendliness, and high-dimensional scalability.

The project includes two examples:

1. Burgers equation (nonlinear wave model in fluid mechanics): Computational domain x∈[-1,1], t∈[0,1], initial condition u(x,0)=-sin(πx), boundary conditions u(-1,t)=u(1,t)=0.
2. Poisson equation (representative of elliptic PDEs): Describes steady-state field distribution; the network needs to satisfy boundary conditions and the Laplacian equals the source term f.

The project uses Keras to build a fully connected network, with spatiotemporal/spatial coordinates as input and solution predictions as output. Key technologies:

• Automatic differentiation: Use TensorFlow GradientTape to compute derivatives of all orders;
• Composite loss function: L_total = L_pde (PDE residual) + L_ic (initial condition error) + L_bc (boundary condition error);
• Training strategy: Adam optimizer, random sampling of collocation points to compute loss.

The project can be extended to multiple fields:

• Fluid mechanics: Navier-Stokes equation solving, turbulence simulation;
• Heat transfer and mass transfer: Heat equation, phase change simulation;
• Solid mechanics: Elastic deformation, fracture mechanics;
• Geophysics and energy: Seismic wave propagation, reservoir simulation;
• Biomedical: Hemodynamics, drug diffusion.

Recommended learning steps:

1. Master PDE classification and classic numerical solution methods;
2. Read the project's Burgers equation example to understand the network structure, loss construction, and training process;
3. Modify hyperparameters, sampling points, etc., and observe result changes;
4. Try to solve PDE problems in your own field;
5. Study Raissi's original papers to understand the theory and latest developments.

PINNs are an important advancement in scientific machine learning, combining the expressive power of neural networks with physical constraints to provide new approaches for complex physical problems. This project concisely covers the core elements of PINNs and is a high-quality resource for getting started. In the future, PINNs are expected to play a role in more scientific and engineering fields, serving as a bridge between data and physical laws.