Physics-Informed Wavelet–Fourier Model for Multiscale Nonlinear PDEs

This repository contains the official implementation of the paper:
"Physics-informed wavelet–Fourier model for multiscale nonlinear partial differential equations"

The proposed PIWF model is a physics-informed unsupervised learning framework that accurately solves multiscale nonlinear PDEs with discontinuities and complex structures. It integrates a wavelet layer for multiscale local feature extraction, a Fourier-branch for global frequency representation, a residual MLP for nonlinear mapping, and a channel attention mechanism for adaptive feature fusion.

📌 Highlights

• 🌊 Wavelet layer – captures sharp gradients and discontinuities via multilevel wavelet series approximation
• 🌍 Fourier branch – maintains global spectral consistency
• 🧠 Residual MLP – enhances expressivity for complex nonlinear mappings
• 🔗 Channel attention – adaptively balances local and global features
• 📉 Physics-informed training – no labeled data required, loss function encodes PDE residuals
• 🔬 Validated on Burgers’ equation, Shallow water equations, and 2D cylinder wake flow
• 📊 Compared with PINN, PIKAN, and FD (finite difference), including ablation studies

Each model folder contains the complete implementation for that method, including training scripts, configuration files, and result visualization.

🚀 Quick Start

Prerequisites

• Python 3.8 or higher
• PyTorch 1.10+
• Other dependencies: numpy, matplotlib, scipy, pyyaml, tqdm

Run an example

To train/test the PIWF model on the 2D cylinder wake flow problem, execute: cd PIWF/cylinder python cylinder_PIWF.py 📊 Reproducing Paper Results  All experimental results reported in the paper can be reproduced by running the provided scripts with default hyperparameters