Hyperparameter Optimization in the Estimation of PDE and Delay-PDE models from data

TL;DR

This paper introduces a Bayesian optimization-based approach for estimating PDE and delay-PDE models from data, enhancing robustness and scope by automatic hyperparameter tuning and integrating time into the estimation process.

Contribution

It presents a novel method combining Bayesian optimization with model estimation, allowing automatic hyperparameter selection and improved modeling of complex physical systems.

Findings

Enhanced robustness in PDE model estimation.

Successful application to synthetic benchmark problems.

Broader modeling scope with automatic hyperparameter tuning.

Abstract

We propose an improved method for estimating partial differential equations and delay partial differential equations from data, using Bayesian optimization and the Bayesian information criterion to automatically find suitable hyperparameters for the method itself or for the equations (such as a time-delay). We show that combining time integration into an established model estimation method increases robustness and yields predictive models. Allowing hyperparameters to be optimized as part of the model estimation results in a wider modelling scope. We demonstrate the method's performance on a number of synthetic benchmark problems of different complexity, representing different classes of physical behaviour. This includes the Allen-Cahn and Cahn-Hilliard models, as well as different reaction-diffusion systems without and with time-delay.