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

TL;DR

The paper tackles the challenge of learning accurate parametric reduced-order models (ROMs) for affine-parametric dynamical systems under limited training data. It develops a Bayesian affine-parametric operator inference (pOpInf) framework that yields predictive uncertainty through a Gaussian posterior over reduced operators, enabling uncertainty-informed acquisition of new training data. Two acquisition functions are proposed: a stability-based probability of ROM instability and a global-uncertainty variance score, which together guide adaptive sampling to regions that improve global ROM performance with a fixed computational budget. Numerical experiments on a heat-diffusion-reaction PDE and a 2D Burgers' equation show that the active-learning approach yields more stable and accurate ROMs with about half as many full-order solves as random sampling, illustrating practical value for digital twin workflows and uncertainty quantification.

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.

Figures (11)

• Figure 1: Solutions to the full-order diffusion-reaction model \ref{['eq:heatfom']} for values of the parameter vector $\vb*{\xi} = (\kappa,\rho)$ at the corners of the parameter domain $\mathcal{P} = [10^{-3}, 10^{-1}] \times [1, 5]$. The initial condition and time domain are the same for all experiments.
• Figure 2: Acquisition values $\varphi(\vb*{\xi})$ (top) and relative $L^2$-norm ROM errors (bottom) after training on $n_p$ adaptively chosen parameter samples for the problem \ref{['eq:heatpde']}.
• Figure 3: Parameter samples selected from a $20\times 20$ grid $\mathcal{G}$ over the parameter domain $\mathcal{P}$ using adaptive sampling (top) and Latin hypercube sampling (bottom) for the problem \ref{['eq:heatpde']}. Results are shown for three trials, each with a different random initial training parameter vector (blue).
• Figure 4: Percentages of the $400$ training parameter candidates where any instabilities are observed from $50$ posterior draws of the probabilistic ROM, as a function of the number of training parameters $n_p$, for the problem \ref{['eq:heatpde']}. The results are averaged and maximized over $50$ trials with different random initial parameter samples.
• Figure 5: Total relative $L^2$ ROM errors $\mathcal{E}_\textrm{total}$ as a function of the number of training parameter samples, aggregated across 50 trials with different initial samples, for the problem \ref{['eq:heatpde']}. The shaded regions show the interquantile range of $5\%$--$95\%$ of trials.

Theorems & Definitions (4)

• Remark 2.1: Dimensionality reduction
• Remark 3.1: Equality of total variance in reduced and original state spaces
• Remark 3.2: Shifted approximation
• Remark 4.1: Compressed Kronecker product