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

• URL: http://arxiv.org/abs/2210.07880v1
• Date: Fri, 14 Oct 2022 15:01:32 GMT
• Title: Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations
• Authors: Alexander New and Benjamin Eng and Andrea C. Timm and Andrew S. Gearhart
• Abstract summary: In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity 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.
• Score: 64.78260098263489
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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