Learning Physics-Informed Neural Networks from First Principles: A Systematic SciML Implementation Library

The GitHub project Physics_Informed_Learning maintained by raj8102018 provides a systematic path to learn Physics-Informed Neural Networks (PINNs) from first principles. The project includes from-scratch implementations of core concepts like PINNs and operator learning, covering classic Partial Differential Equation (PDE) cases such as diffusion equations, Burgers' equations, and Poisson equations. It uses multiple frameworks like PyTorch, JAX, and DeepXDE for implementation, balancing theoretical understanding and engineering applications (e.g., Terzaghi consolidation in geotechnical engineering), making it a high-quality learning resource in the field of Scientific Machine Learning (SciML).

Physics-Informed Machine Learning (SciML) is an interdisciplinary field that intersects machine learning, scientific computing, and numerical methods. The core idea of PINNs is to encode physical laws into the neural network's loss function: compute PDE residuals via automatic differentiation, allowing the model to fit observed data while satisfying physical constraints, thus solving the problem of physical consistency in data-sparse scenarios. This project is positioned as a systematic learning library rather than a simple reproduction, aiming for framework-agnostic deep understanding.

The project directory is designed according to the learning path, progressing systematically from basic PDE cases to engineering applications. Each case is implemented using multiple frameworks like PyTorch, JAX, and DeepXDE, stripping away framework-specific syntactic sugar to focus directly on core mathematical concepts. Core methods include: PDE residual calculation (high-order derivatives via automatic differentiation), boundary/initial condition handling (penalizing deviations in the loss function), loss balancing strategies (weighting data loss and physical loss), etc.

The project covers several key cases: 1. Diffusion equation (PINN introduction, including residual, boundary/initial condition construction); 2. Burgers' equation (nonlinear PDE challenge, handling convection terms via automatic differentiation); 3. Inverse diffusion problem (inferring unknown diffusion coefficients, demonstrating inverse problem solving capability); 4. 2D Poisson equation (handling high-dimensional spatial domains); 5. Wave equation (special handling of second-order time derivatives); 6. Terzaghi consolidation (geotechnical engineering application, verifying the engineering value of PINNs).

The project provides a step-by-step learning path (from basics to engineering applications) with handwritten notes recording implementation thoughts, making it an excellent resource for getting started with SciML. SciML represents an important direction of integration between AI and science; through the two-way interaction of physics-inspired AI and AI-driven scientific discovery, it reshapes the way complex problems are solved. The systematic organization and framework-agnostic strategy of this project are of reference value to both machine learning researchers and engineering practitioners.

Future attention can be paid to cutting-edge directions in SciML: training optimization (spectral bias resolution, adaptive collocation points, loss balance improvement); operator learning (DeepONet, Fourier neural operators, Transformer operators); application expansion (leachate prediction in landfills, pollutant transport modeling, geotechnical engineering digital twins), etc. These directions promote the transition of PINNs from theory to practical applications.