Adversarial Sampling for Solving Differential Equations with Neural Networks

TL;DR

This paper introduces an adversarial sampling method for neural network-based differential equation solvers, which adaptively selects points to maximize residuals, leading to improved accuracy over traditional sampling methods.

Contribution

The paper proposes a novel adversarial sampling scheme and architecture that enhances neural differential equation solvers by focusing on challenging points.

Findings

Adversarial sampling outperforms uniform sampling in solving differential equations.

The proposed method achieves higher accuracy on benchmark problems.

The approach effectively identifies critical points that improve solution quality.

Abstract

Neural network-based methods for solving differential equations have been gaining traction. They work by improving the differential equation residuals of a neural network on a sample of points in each iteration. However, most of them employ standard sampling schemes like uniform or perturbing equally spaced points. We present a novel sampling scheme which samples points adversarially to maximize the loss of the current solution estimate. A sampler architecture is described along with the loss terms used for training. Finally, we demonstrate that this scheme outperforms pre-existing schemes by comparing both on a number of problems.