Rank-Minimizing and Structured Model Inference

TL;DR

This paper presents a novel method for inferring low-order, structured models from data that incorporate physical principles, using rank minimization of Sylvester equation solutions to improve model simplicity and accuracy.

Contribution

It introduces a rank-minimization approach for structured model inference that automatically reduces model complexity while preserving physical insights.

Findings

Models have significantly fewer degrees of freedom than comparable methods.

The approach maintains high prediction accuracy with low-order models.

Numerical experiments validate the effectiveness of the method.

Abstract

While extracting information from data with machine learning plays an increasingly important role, physical laws and other first principles continue to provide critical insights about systems and processes of interest in science and engineering. This work introduces a method that infers models from data with physical insights encoded in the form of structure and that minimizes the model order so that the training data are fitted well while redundant degrees of freedom without conditions and sufficient data to fix them are automatically eliminated. The models are formulated via solution matrices of specific instances of generalized Sylvester equations that enforce interpolation of the training data and relate the model order to the rank of the solution matrices. The proposed method numerically solves the Sylvester equations for minimal-rank solutions and so obtains models of low order. Numerical experiments demonstrate that the combination of structure preservation and rank minimization leads to accurate models with orders of magnitude fewer degrees of freedom than models of comparable prediction quality that are learned with structure preservation alone.