Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations

TL;DR

This paper evaluates the performance of physics-informed neural networks (PINNs) on complex coupled ODE benchmarks with tunable difficulty, revealing limitations in their ability to solve high-complexity problems despite advanced training methods.

Contribution

It introduces tunable complexity benchmarks for PINNs and systematically analyzes their failure modes as problem difficulty increases.

Findings

PINNs struggle with high-complexity coupled ODEs.

Increasing problem complexity leads to solution inaccuracies.

Limitations are linked to network capacity, conditioning, and local curvature.

Abstract

In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs). We focus on a pair of benchmarks: discretized partial differential equations and harmonic oscillators, each of which has a tunable parameter that controls its complexity. Even by varying network architecture and applying a state-of-the-art training method that accounts for "difficult" training regions, we show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity -- the number of equations and the size of time domain -- increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.