Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architectures
This addresses the computational inefficiency of training-based methods for physics-informed regression, though it appears incremental as it builds on existing optimization and Taylor series concepts.
The authors tackled physics-informed regression by proposing a method that computes predictions directly via constrained optimization using Taylor series expansions, eliminating the need for training loops. Their approach achieved competitive accuracy on a reaction-diffusion system benchmark compared to neural networks while being robust to sampling layout changes.
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.