Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

Systems exhibiting multiscale, spatiotemporal chaos – the prime example being turbulence – are difficult to predict due to their large number of degrees of freedom. Often, we know the governing equations (i.e., full order model (FOM)) for these systems, but directly solving them can be computationally prohibitive. Therefore, high-fidelity, reduced-order models (ROMs) are needed for multi-query tasks such as controller design for these complex systems. In this work, we build a hybrid ROM that uses physics (knowledge of the FOM) to inform a purely data-driven ROM called Data-driven Manifold Dynamics (DManD) [1]. DManD uses a neural ordinary differential equation (NODE) to learn the vector field on a low-dimensional latent space discovered with an undercomplete autoencoder. We incorporate physics into DManD by modifying a method called Manifold Galerkin (MG) [2]. MG provides an expression for the vector field on the latent space in terms of the vector field on the full space (i.e., FOM). In our approach, denoted physics-informed DManD (PI-DManD) [3], this physics-based vector field is either corrected with dynamic data using a NODE, or used as a Bayesian prior for training a NODE. We evaluate this approach with two systems exhibiting spatiotemporal chaos, the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations. Both PI-DManD methods track the true solution much better than DManD. At one Lyapunov time, the ensemble-averaged, normalized error is 0.05 for PI-DManD and 0.1 for DManD. Lastly, we find that PI-DManD (particularly the Bayesian version) substantially outperforms DManD in tracking performance even in scenarios of scarce data and inexact physics (e.g., uncertainty in FOM parameters).



[1] Linot et al. (2020) https://doi.org/10.1103/PhysRevE.101.062209

[2] Lee et al. (2020) https://doi.org/10.1016/j.jcp.2019.108973

[3] Guo and Graham ArXiv (2025)