Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations
This work addresses the reliability of PINNs for solving complex ODEs, which is crucial for researchers and practitioners in scientific computing, but it is incremental as it builds on existing benchmarks and methods.
The paper tackled the problem of evaluating physics-informed neural networks (PINNs) on increasingly complex coupled ordinary differential equations (ODEs) using tunable benchmarks, and found that PINNs fail to produce correct solutions as complexity increases, despite varying architectures and advanced training methods.
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.