Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architectures

TL;DR

This paper introduces a direct, optimization-based approach for physics-informed regression that computes predictions and derivatives without training, offering competitive accuracy and robustness compared to neural networks.

Contribution

The proposed method directly computes physics-informed predictions via constrained optimization, eliminating training loops and bespoke architectures.

Findings

Competitive accuracy on reaction-diffusion systems

No need for training or re-training

Robust to sampling layout changes

Abstract

We revisit the problem of physics-informed regression, and propose a method that directly computes the state at the prediction point, simultaneously with the derivative and curvature information of the existing samples. We frame each prediction as a constrained optimisation problem, leveraging multivariate Taylor series expansions and explicitly enforcing physical laws. Each individual query can be processed with low computational cost without any pre- or re-training, in contrast to global function approximator-based solutions such as neural networks. Our comparative benchmarks on a reaction-diffusion system show competitive predictive accuracy relative to a neural network-based solution, while completely eliminating the need for long training loops, and remaining robust to changes in the sampling layout.