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

TL;DR

This work develops a solver-free, data-driven framework for learning continuous-time stochastic reduced-order models (ROMs) of parametrized dynamical systems by amortized stochastic variational inference. It jointly learns a probabilistic autoencoder and a latent SDE, using a reparametrized ELBO to avoid expensive forward solvers and to quantify prediction uncertainty; physics-informed priors can be incorporated to guide the latent dynamics. The method is amortized over time partitions with a continuous-time latent representation via a deep kernel, enabling scalable training that scales with the number of samples $R$, window size $M$, and time samples $L$, while remaining independent of dataset size and stiffness. Results on reaction-diffusion, forced Burgers’ equation, and large-scale flow control demonstrate strong generalization across unseen parameters and forcings, with substantial reductions in training time compared to neural ODE/SDE baselines and clear uncertainty quantification linked to high-error regions. Overall, the approach bridges physics-informed and data-driven ROMs, offering a scalable, uncertainty-aware tool for robust prediction and decision-making in complex dynamical 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.

Figures (11)

• Figure 1: Graphical overview of the proposed stochastic ROM approach. The ROM, which we train by stochastic variational inference, consists of three modules: a probabilistic encoder, a latent SDE, and a probabilistic decoder. Making a prediction with the ROM starts with parameters $\mu$, forcing $f(t)$, and initial condition $u_0$. We encode the initial condition to obtain the corresponding latent $z_0$. We then solve the latent SDE to obtain realizations of the trajectory $z(t)$. Decoding the sampled trajectories yields the stochastic ROM prediction of the FOM QoI, $u(t)$, that can be postprocessed for the mean prediction and statistical error bars.
• Figure 2: Reaction-diffusion: error evolution and predictions over time. The top row illustrates prediction error on the test interval for noiseless (left) and noisy (right) data. The bottom half shows the true solution, prediction mean, absolute error, and prediction standard deviation. See movies S1-S2.
• Figure 3: Burgers' equation: solution, error, & prediction statistics for a test trajectory. See movie S3.
• Figure 4: Burgers' equation: error distribution on test set over time for $d=5$.
• Figure 5: Fluid flow with control: solution, error, & prediction statistics for a test trajectory. Only the velocity field on the left half of the domain is shown. See movie S4.