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

2025-12-30

AI Summary

This work proposes a probabilistic reduced-order modeling (ROM) framework that integrates Bayesian operator inference with active learning to address the challenge of efficiently selecting training parameters under limited computational budgets while ensuring both global accuracy and stability. By introducing Bayesian linear regression into parametric operator inference for the first time, the method enables rigorous quantification of predictive uncertainty. An adaptive sampling strategy is further developed, which dynamically selects the most informative parameter samples based on this uncertainty. Numerical experiments on multiple nonlinear parametrized partial differential equation systems demonstrate that, compared to random sampling, the proposed approach significantly enhances ROM accuracy and stability at equivalent computational cost.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• active learning
• reduced-order models
• parametric differential systems
• Bayesian operator inference
• adaptive sampling

Innovation

• active learning
• Bayesian operator inference
• reduced-order modeling
• adaptive sampling
• parametric dynamical systems