ELM-FBPINN: efficient finite-basis physics-informed neural networks

TL;DR

This work significantly accelerate the training of FBPINNs by linearising their underlying optimisation problem by employing extreme learning machines (ELMs) as their subdomain networks and showing that this turns the FBPINN optimisation problem into one of solving a linear system or least-squares problem.

Abstract

Physics Informed Neural Networks (PINNs) offer several advantages when compared to traditional numerical methods for solving PDEs, such as being a mesh-free approach and being easily extendable to solving inverse problems. One promising approach for allowing PINNs to scale to multi-scale problems is to combine them with domain decomposition; for example, finite basis physics-informed neural networks (FBPINNs) replace the global PINN network with many localised networks which are summed together to approximate the solution. In this work, we significantly accelerate the training of FBPINNs by linearising their underlying optimisation problem. We achieve this by employing extreme learning machines (ELMs) as their subdomain networks and showing that this turns the FBPINN optimisation problem into one of solving a linear system or least-squares problem. We test our workflow in a preliminary fashion by using it to solve an illustrative 1D problem.

Figures (3)

• Figure 1: Left: Accuracy comparison for 1D damped harmonic oscillator. Right: Final $L_1$ loss values for each method.
• Figure 2: Time (s) vs loss (log) for each method.
• Figure 3: The condition number (log) vs network size for PINN and FBPINN.