Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference

TL;DR

This paper introduces an active learning framework using Bayesian operator inference to efficiently select training data, significantly improving the stability and accuracy of reduced-order models for parametric dynamical systems.

Contribution

It develops a probabilistic, Bayesian approach to active data selection for data-driven ROMs, enhancing model stability and accuracy over random sampling methods.

Findings

Adaptive sampling improves ROM stability and accuracy.

Bayesian operator inference quantifies prediction uncertainty.

Proposed method outperforms random sampling in numerical experiments.

Abstract

This work develops an active learning framework to intelligently enrich data-driven reduced-order models (ROMs) of parametric dynamical systems, which can serve as the foundation of virtual assets in a digital twin. Data-driven ROMs are explainable, computationally efficient scientific machine learning models that aim to preserve the underlying physics of complex dynamical simulations. Since the quality of data-driven ROMs is sensitive to the quality of the limited training data, we seek to identify training parameters for which using the associated training data results in the best possible parametric ROM. Our approach uses the operator inference methodology, a regression-based strategy which can be tailored to particular parametric structure for a large class of problems. We establish a probabilistic version of parametric operator inference, casting the learning problem as a Bayesian linear regression. Prediction uncertainties stemming from the resulting probabilistic ROM solutions are used to design a sequential adaptive sampling scheme to select new training parameter vectors that promote ROM stability and accuracy globally in the parameter domain. We conduct numerical experiments for several nonlinear parametric systems of partial differential equations and compare the results to ROMs trained on random parameter samples. The results demonstrate that the proposed adaptive sampling strategy consistently yields more stable and accurate ROMs than random sampling does under the same computational budget.