An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks

TL;DR

This paper presents RSmote, a novel adaptive sampling method for Physics-Informed Neural Networks that improves accuracy and efficiency by focusing on high residual regions and reducing memory consumption in high-dimensional problems.

Contribution

Introduction of RSmote, a residual-based adaptive sampling technique that enhances PINN performance and resource efficiency compared to existing methods.

Findings

RSmote matches or exceeds RAD accuracy in various problems.

RSmote significantly reduces memory usage in high-dimensional cases.

RSmote improves computational efficiency for complex PDEs.

Abstract

This paper introduces Residual-based Smote (RSmote), an innovative local adaptive sampling technique tailored to improve the performance of Physics-Informed Neural Networks (PINNs) through imbalanced learning strategies. Traditional residual-based adaptive sampling methods, while effective in enhancing PINN accuracy, often struggle with efficiency and high memory consumption, particularly in high-dimensional problems. RSmote addresses these challenges by targeting regions with high residuals and employing oversampling techniques from imbalanced learning to refine the sampling process. Our approach is underpinned by a rigorous theoretical analysis that supports the effectiveness of RSmote in managing computational resources more efficiently. Through extensive evaluations, we benchmark RSmote against the state-of-the-art Residual-based Adaptive Distribution (RAD) method across a variety of dimensions and differential equations. The results demonstrate that RSmote not only achieves or exceeds the accuracy of RAD but also significantly reduces memory usage, making it particularly advantageous in high-dimensional scenarios. These contributions position RSmote as a robust and resource-efficient solution for solving complex partial differential equations, especially when computational constraints are a critical consideration.