Stable reduced models improve long-term PDE simulations across applications
SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs
Machine LearningArtificial Intelligence
Summary
Simulating complex processes like waves or fluid flow using math equations takes a lot of computing power, especially when predicting far into the future. The authors present SMORE, a new method that simplifies these equations without losing stability, making long-term predictions more reliable and easier to compute. Their technique learns a simple and stable representation of how the system evolves over time, which helps avoid errors growing out of control. They tested SMORE on several scientific problems and showed it works better or as well as existing methods while being more stable for long predictions.
What this means in practice
- •For simulation engineers: Run faster, more stable long-term simulations of physical systems governed by PDEs with fewer computational resources.
- •For control systems developers: Use interpretable, stable latent models to improve data assimilation and optimal control tasks involving time-dependent PDEs.
Authors
Yangyuan Li, Weichao Li, Shaowu Pan
Abstract
High-fidelity simulations of time-dependent partial differential equations (PDEs) are computationally expensive, motivating data-driven reduced-order surrogates for many-query tasks such as uncertainty quantification, design optimization, data assimilation, and optimal control. However, existing surrogate models often exhibit poor temporal stability, which can lead to unstable rollouts and exploding gradients during backpropagation, especially in multistep long-horizon forecasting. To address this, we propose SMORE, a mesh-agnostic framework for model order reduction of time-dependent PDEs. Its latent dynamics are trained with Lyapunov-guided stability regularization, which promotes stable long-horizon rollouts. We provide theoretical guarantees under the stated structural assumptions. Beyond forecasting PDE evolution, the learned latent dynamics, which are interpretable and linear or linear-quadratic, could bring benefits for downstream tasks such as data assimilation and optimal control. Moreover, our framework is capable of predicting continuous PDE solution fields from sparse measurements of the initial condition. We evaluate SMORE on a range of problems, including wave propagation, the Navier-Stokes equations, and the shallow water equations. Our results show that it improves long-horizon rollout generalization and empirical robustness, and achieves competitive accuracy at comparable parameter budgets relative to competitive baselines including DINo, FNO, CNO, and Transolver.