Efficient data-driven regression for reduced-order modeling of spatial pattern formation

TL;DR

This paper introduces a non-intrusive, data-driven regression method for creating efficient reduced-order models of reaction-diffusion systems that form spatial patterns, enabling accurate predictions without detailed physical knowledge.

Contribution

It develops a polynomial-based regression approach using POD for low-cost, accurate ROMs applicable to complex nonlinear pattern-forming systems without requiring physical equations.

Findings

Higher-order surrogate models improve prediction accuracy.

The method maintains low computational cost.

Effective for classical pattern-forming systems like Schnakenberg and Mimura models.

Abstract

We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically accurate numerical simulations. The method can be applied to general nonlinear systems through the use of polynomial model form, while not requiring knowledge of the underlying physical model, governing equations, or numerical solvers. The process of learning ROMs is posed as a low-cost least-squares problem in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical pattern-forming systems--including the Schnakenberg and Mimura--Tsujikawa models--demonstrate that higher-order surrogate models significantly improve prediction accuracy while maintaining low computational cost. The proposed method provides a flexible, non-intrusive model reduction framework, well suited for the analysis of complex spatio-temporal pattern formation phenomena.