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

arXiv: https://arxiv.org/abs/2601.00038 · HTML: https://mlsanity.com/paper/2601.00038/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 80 references that are in the map.
- First posted on arXiv 30 Dec 2025.

## Builds on (most cited first, 23 of 23)

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Bayesian operator inference for data-driven reduced-order modeling](https://mlsanity.com/paper/2204.10829/) (2022)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems](https://mlsanity.com/paper/2404.05040/) (2024)
- [Progressive construction of a parametric reduced-order model for PDE-constrained optimization](https://mlsanity.com/paper/1407.7618/) (2014)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Stability Domains for Quadratic-Bilinear Reduced-Order Models](https://mlsanity.com/paper/2009.02769/) (2020)
- [Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression](https://mlsanity.com/paper/2302.08216/) (2023)
- [A Quadratic Manifold for Model Order Reduction of Nonlinear Structural Dynamics](https://mlsanity.com/paper/1610.09902/) (2016)
- [Grassmannian diffusion maps based surrogate modeling via geometric harmonics](https://mlsanity.com/paper/2109.13805/) (2021)
- [Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation](https://mlsanity.com/paper/2210.14764/) (2022)
- [Block-structured Operator Inference for coupled multiphysics model reduction](https://mlsanity.com/paper/2511.05389/) (2025)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [Active-learning-based non-intrusive Model Order Reduction](https://mlsanity.com/paper/2204.08523/) (2022)
- [Physically consistent predictive reduced-order modeling by enhancing Operator Inference with state constraints](https://mlsanity.com/paper/2502.03672/) (2025)
- [Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction](https://mlsanity.com/paper/2507.20463/) (2025)

## Abstract (arXiv)

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.