Adversarial Learning for Neural PDE Solvers with Sparse Data

TL;DR

This paper introduces SMART, a universal data augmentation strategy for neural PDE solvers that enhances robustness and accuracy in data-scarce environments by focusing on model weaknesses, validated through theory and experiments.

Contribution

The study presents SMART, a novel augmentation method that improves neural PDE solver performance under limited data by targeting model weaknesses, without relying on strong physical assumptions.

Findings

SMART reduces generalization error in neural PDEs.

Significant accuracy improvements demonstrated across various PDE scenarios.

Method validated through theoretical analysis and extensive experiments.

Abstract

Neural network solvers for partial differential equations (PDEs) have made significant progress, yet they continue to face challenges related to data scarcity and model robustness. Traditional data augmentation methods, which leverage symmetry or invariance, impose strong assumptions on physical systems that often do not hold in dynamic and complex real-world applications. To address this research gap, this study introduces a universal learning strategy for neural network PDEs, named Systematic Model Augmentation for Robust Training (SMART). By focusing on challenging and improving the model's weaknesses, SMART reduces generalization error during training under data-scarce conditions, leading to significant improvements in prediction accuracy across various PDE scenarios. The effectiveness of the proposed method is demonstrated through both theoretical analysis and extensive experimentation. The code will be available.