Building Physics-Informed Neural Networks from Scratch: A PINN Learning Path and Implementation Guide

This article introduces the open-source project DoZeroAsPINN, which provides a structured path for learning Physics-Informed Neural Networks (PINN) from scratch. PINN integrates deep learning and physical laws, offering a new paradigm for solving scientific computing and engineering problems. Using a progressive approach from basic concepts to practical implementation, the project helps learners gain an in-depth understanding of PINN's internal mechanisms.

Limitations of Traditional Scientific Computing

Scientific and engineering problems are often described using partial differential equations (PDEs), but traditional numerical methods (finite difference, finite element) face challenges such as grid dependence, curse of dimensionality, and data scarcity.

Limitations of Pure Data-Driven Methods

Pure deep learning models are "black boxes" that do not guarantee physical constraints and tend to produce unreasonable predictions in areas not covered by data.

The Integration Approach of PINN

PINN embeds physical laws (in PDE form) as soft constraints into the loss function, enabling the model to both fit data and satisfy physical rules, achieving a "data + physics" dual-driven approach.

Mathematical Foundations

• General form of PDE: ν_t + N[u] = 0 (u is the field variable, N is the differential operator)
• Neural network approximation: Use u_θ(t,x) to approximate the solution, where θ is the parameter
• Automatic differentiation: Compute derivatives of various orders (time, space) to construct PDE residuals
• Composite loss function: L(θ)=L_data+L_PDE+L_BC+L_IC (data fitting, PDE residual, boundary/initial condition losses)

Learning Path

Phase 1: Deep learning basics (neural network structure, backpropagation, automatic differentiation) Phase 2: Differential equation basics (ODE/PDE concepts, numerical methods) Phase 3: PINN core (solution representation, derivative calculation, loss construction) Phase 4: Advanced topics (complex geometry, inverse problems, high-dimensional problems) Phase 5: Practical applications (fluid mechanics, solid mechanics, etc.)

Training Difficulties and Convergence Issues

Competition between composite loss terms leads to unstable optimization; solutions: adaptive loss weights, curriculum learning, L-BFGS optimizer.

High-Frequency and Multi-Scale Problems

Neural networks have a spectral bias towards low frequencies; solutions: Fourier feature embedding, multi-scale architecture, domain decomposition.

Stiff Problems and Long-Term Time Domain Integration

Error accumulation issues; solutions: sequence-to-sequence learning, autoregressive PINN, combining with traditional integrators.

• Computational Fluid Dynamics (CFD)：Solve Navier-Stokes equations, handle complex geometry and inverse problems
• Solid Mechanics: Linear elasticity, nonlinear elasticity, etc., infer material properties
• Heat Conduction and Mass Transfer: Heat conduction equations, handle scenarios like unknown heat sources
• Earth Sciences: Complex problems like seismic wave propagation, groundwater flow
• Biomedical Engineering: Multi-physics coupling problems like hemodynamics, drug diffusion

Feature	Traditional Numerical Methods	PINN
Grid Requirement	Requires discrete grid	Mesh-free
Dimensional Scalability	Curse of dimensionality	Relatively better
Inverse Problem Handling	Requires specialized framework	Naturally supported
Data Integration	Difficult	Naturally integrated
Computational Efficiency	Mature and efficient	High training cost
Interpretability	High	Relatively low
Convergence Guarantee	Theoretical guarantee	Empirical

PINN and traditional methods are complementary: PINN has obvious advantages in scenarios with scarce data, complex geometry, and inverse problems; traditional methods are still needed for high-precision and convergence-guaranteed scenarios.

Mathematical Foundation Preparation

Linear algebra, multivariable calculus, ODE/PDE theory, basics of numerical methods

Programming Skills

Python scientific computing (NumPy/SciPy), deep learning frameworks (PyTorch/TensorFlow), Git basics

Practical Suggestions

1. Start with 1D PINN implementation
2. Visualize predictions, residuals, and training processes
3. Validate by comparing with analytical solutions/traditional numerical solutions
4. Follow the latest papers (arXiv)
5. Participate in open-source projects and community exchanges

The DoZeroAsPINN project provides a structured entry path for PINN learners. PINN is a cutting-edge direction in the fusion of scientific computing and machine learning, and its dual-driven paradigm changes the way scientific problems are solved. For engineering practitioners: PINN can leverage physical priors when data is incomplete; for researchers: there are open issues such as training stability and multi-scale modeling. With the popularization of computing resources and the maturity of frameworks, the threshold for PINN is lowering, and open-source resources will promote the development of scientific machine learning.