Neuroevolution of Physics-Informed Neural Nets: Benchmark Problems and Comparative Results

TL;DR

This paper introduces benchmark problems for physics-informed neural networks (PINNs) and demonstrates that neuroevolution algorithms can outperform traditional gradient descent methods in training PINNs, leading to better physical law compliance.

Contribution

The paper presents five benchmark problems for PINNs and compares neuroevolution algorithms with gradient descent, showing neuroevolution's superior global search capabilities and improved training outcomes.

Findings

Neuroevolution algorithms outperform gradient descent in PINN training.

Benchmark problems facilitate development of novel neuroevolution methods.

Implementing neuroevolution with JAX significantly speeds up training.

Abstract

The potential of learned models for fundamental scientific research and discovery is drawing increasing attention worldwide. Physics-informed neural networks (PINNs), where the loss function directly embeds governing equations of scientific phenomena, is one of the key techniques at the forefront of recent advances. PINNs are typically trained using stochastic gradient descent methods, akin to their deep learning counterparts. However, analysis in this paper shows that PINNs' unique loss formulations lead to a high degree of complexity and ruggedness that may not be conducive for gradient descent. Unlike in standard deep learning, PINN training requires globally optimum parameter values that satisfy physical laws as closely as possible. Spurious local optimum, indicative of erroneous physics, must be avoided. Hence, neuroevolution algorithms, with their superior global search capacity, may be a better choice for PINNs relative to gradient descent methods. Here, we propose a set of five benchmark problems, with open-source codes, spanning diverse physical phenomena for novel neuroevolution algorithm development. Using this, we compare two neuroevolution algorithms against the commonly used stochastic gradient descent, and our baseline results support the claim that neuroevolution can surpass gradient descent, ensuring better physics compliance in the predicted outputs. %Furthermore, implementing neuroevolution with JAX leads to orders of magnitude speedup relative to standard implementations.