Projection-based model reduction of dynamical systems using space-time subspace and machine learning

TL;DR

This paper introduces a space-time model reduction technique for high-dimensional spatiotemporal systems, combining tensor decompositions with machine learning to create efficient parametric surrogate models for engineering applications.

Contribution

It develops a novel low-dimensional space-time basis using tensor decompositions and compares multiple machine learning methods for parameter-to-coefficient mapping, emphasizing efficiency in data-scarce scenarios.

Findings

The proposed method achieves high accuracy with reduced computational cost.

Simpler machine learning models outperform more complex ones in limited data settings.

The approach is validated on three engineering problems with promising results.

Abstract

This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional spatiotemporal output quantities of interest, such as pressure, temperature and displacement fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using space-time bases combining with machine learning methods. In particular, the space-time solutions are sought in a low-dimensional space-time linear trial subspace that can be obtained by computing tensor decompositions of usual state-snapshots data. The mapping between the input parameters and the basis expansion coefficients (or generalized coordinates) is approximated using four different machine learning techniques: multivariate polynomial regression, k-nearest-neighbors, random forest and neural network. The relative costs and effectiveness of the four machine learning techniques are explored through three engineering problems: steady heat conduction, unsteady heat conduction and unsteady advective-diffusive-reactive system. Numerical results demonstrate that the proposed method performs well in terms of both accuracy and computational cost, and highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models. In addition, simpler machine learning techniques are seen to perform better than more elaborate ones.