Analysis and Practice of PDE Solving Methods Based on Fourier Neural Operator

Type: Undergraduate Thesis Completion Date: June 2024

Background

Partial Differential Equations (PDEs) are fundamental mathematical tools for describing natural phenomena and engineering problems, widely applied in fluid dynamics, electromagnetism, quantum mechanics, and more. While traditional numerical methods (e.g., finite element, finite difference) are mature, they incur high computational costs when dealing with high-dimensional problems, complex boundary conditions, or scenarios requiring numerous solutions.

In recent years, deep learning has brought new possibilities to scientific computing. The Fourier Neural Operator (FNO), as a novel operator learning method, can learn solution operators of PDEs in function spaces, achieving solving speeds several orders of magnitude faster than traditional methods.

Core Content

1. Theoretical Analysis of Fourier Neural Operator

In-depth study of FNO's mathematical principles:

Operator Learning Framework: Treating PDE solving as learning mappings from parameter space to solution space
Fourier Layer Design: Performing global convolution in frequency domain to capture long-range dependencies
Resolution Invariance: Can solve on grids of different resolutions after training
Universal Approximation Theorem: Theoretical guarantee that FNO can approximate arbitrary continuous operators

2. Method Implementation and Optimization

Implemented a complete FNO framework:

Architecture Design: Multi-layer Fourier layers + activation functions + residual connections
Frequency Truncation Strategy: Retaining low-frequency modes to reduce computational complexity
Training Techniques: Data augmentation, learning rate scheduling, regularization methods
Parallel Acceleration: Leveraging FFT efficiency and GPU parallel computing

3. Solving Practice on Typical Equations

Validated method effectiveness on multiple classic PDEs:

Burgers Equation

One-dimensional nonlinear advection-diffusion equation
Testing generalization under varying viscosity coefficients
Relative error < 1%, inference speed improved by 1000+ times

Navier-Stokes Equation

Two-dimensional incompressible fluid dynamics equation
Learning long-term evolution of turbulent patterns
Maintaining good performance at unseen Reynolds numbers

Darcy Flow

Flow in porous media
Handling high-dimensional parameter spaces (permeability fields)
Achieving fast parameterized solutions

Experimental Results

Accuracy Comparison

Compared with traditional numerical methods:

Burgers Equation: L2 error 0.8% vs traditional 0.5%
Navier-Stokes: Error < 3% at time T=10
Darcy Flow: Correlation coefficient > 0.98

Speed Advantage

Post-training Inference: < 0.1 seconds per solution (GPU)
Traditional Methods: Minutes to hours required
Application Scenarios: Parametric studies requiring numerous solutions, real-time simulation

Generalization Capability

Resolution generalization: Solving at 2-4x training grid resolution
Parameter generalization: Maintaining reasonable accuracy beyond training parameter ranges
Equation family generalization: Quickly adapting to related equations with fine-tuning

Advantages and Limitations

Advantages

Fast Solving: Extremely fast inference speed after training, suitable for high-throughput computing
End-to-End Learning: Automatically learns features without manual discretization schemes
Resolution Flexibility: Can solve on different grids
Parallel-Friendly: Naturally suitable for GPU acceleration

Limitations

Data Dependency: Requires large amounts of training data (from traditional solvers or experiments)
Accuracy Gap: Still inferior to traditional methods for high-precision requirements
Interpretability: Black-box nature makes error analysis difficult
Boundary Conditions: Complex boundary condition handling needs improvement

Technical Implementation

Code Framework

Complete training and inference pipeline based on PyTorch
Fast Fourier Transform using torch.fft
Unified interface supporting 1D, 2D, 3D problems

Data Generation

Training data generated using traditional solvers (FEniCS, FDM)
Latin hypercube sampling ensures parameter space coverage
Data augmentation: rotation, translation, scaling

Visualization Analysis

Spatiotemporal evolution visualization of solution fields
Error distribution heatmaps
Spectral analysis: comparing learned frequency domain weights

Future Directions

Physics Embedding: Incorporating physical constraints (conservation laws, symmetries) into networks
Multiscale Modeling: Combining wavelet transforms for multiscale problems
Inverse Problem Solving: For parameter inversion and data assimilation
Real Applications: Deployment in weather forecasting and engineering optimization

Research Significance

This thesis systematically analyzes the application of Fourier Neural Operator in PDE solving:

Theoretical Level: Deep understanding of the mathematical foundations of operator learning
Practical Level: Validated method effectiveness on multiple benchmark problems
Application Level: Provides new tools for data-driven scientific computing

This research demonstrates the great potential of combining deep learning with traditional scientific computing, contributing methodologies and practical experience to the "AI for Science" field.

Keywords: Fourier Neural Operator | Partial Differential Equations | Operator Learning | Scientific Computing | Deep Learning

Undergraduate Thesis: Analysis and Practice of PDE Solving Methods Based on Fourier Neural Operator