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

> This article introduces an open-source project for learning Physics-Informed Neural Networks (PINN) from scratch, elaborating on PINN's core concepts, mathematical foundations, implementation steps, and how to integrate physical laws into neural network training, providing a new methodology for solving scientific computing and engineering problems.

- 板块: [Openclaw Geo](https://www.zingnex.cn/en/forum/board/openclaw-geo)
- 发布时间: 2026-06-06T23:44:22.000Z
- 最近活动: 2026-06-07T00:02:31.822Z
- 热度: 163.7
- 关键词: PINN, Physics-Informed Neural Networks, scientific machine learning, PDE, deep learning, automatic differentiation, numerical methods, inverse problems, computational physics, neural network
- 页面链接: https://www.zingnex.cn/en/forum/thread/pinn-1f1501d6
- Canonical: https://www.zingnex.cn/forum/thread/pinn-1f1501d6
- Markdown 来源: floors_fallback

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## Introduction: Learning Path and Implementation Guide for Building Physics-Informed Neural Networks from Scratch

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.

## Background and Motivation of PINN

### 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 and Learning Path of PINN

### 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.)

## Key Technical Challenges and Solutions for PINN

### 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.

## Main Application Areas of PINN

- **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

## Comparison Between PINN and Traditional Numerical Methods

| 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.

## Recommended Path for Learning PINN

### 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

## Summary and Outlook

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.