Data-driven stochastic reduced-order modeling of parametrized dynamical systems

TL;DR

This paper presents a data-driven stochastic reduced-order modeling framework that efficiently learns probabilistic models for parametrized dynamical systems, enabling uncertainty quantification and generalization across conditions.

Contribution

The authors introduce a novel amortized variational inference approach for learning continuous-time stochastic ROMs that generalize across parameters and incorporate physics-informed priors.

Findings

Excellent generalization to unseen parameters and forcings

Significant computational efficiency over existing methods

Effective uncertainty quantification in complex systems

Abstract

Modeling complex dynamical systems under varying conditions is computationally intensive, often rendering high-fidelity simulations intractable. Although reduced-order models (ROMs) offer a promising solution, current methods often struggle with stochastic dynamics and fail to quantify prediction uncertainty, limiting their utility in robust decision-making contexts. To address these challenges, we introduce a data-driven framework for learning continuous-time stochastic ROMs that generalize across parameter spaces and forcing conditions. Our approach, based on amortized stochastic variational inference, leverages a reparametrization trick for Markov Gaussian processes to eliminate the need for computationally expensive forward solvers during training. This enables us to jointly learn a probabilistic autoencoder and stochastic differential equations governing the latent dynamics, at a computational cost that is independent of the dataset size and system stiffness. Additionally, our approach offers the flexibility of incorporating physics-informed priors if available. Numerical studies are presented for three challenging test problems, where we demonstrate excellent generalization to unseen parameter combinations and forcings, and significant efficiency gains compared to existing approaches.