SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs

2026-09-27

AI Summary

This study addresses the instability and gradient explosion issues inherent in long-term predictions of surrogate models for time-varying partial differential equations (PDEs) by proposing a mesh-independent reduced-order modeling framework. The method constructs interpretable linear latent dynamics through data-driven model order reduction and incorporates Lyapunov stability regularization to constrain the training process, thereby providing rigorous theoretical guarantees for long-horizon evolution and enabling continuous field prediction from sparse initial conditions. Experimental evaluations on the wave equation and Navier-Stokes equations demonstrate that the proposed framework significantly enhances long-term generalization capability and robustness while achieving accuracy comparable to mainstream baseline methods.

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.

Problem

• Time-dependent PDEs
• Model order reduction
• Surrogate models
• Temporal stability
• Long-horizon forecasting

Innovation

• Mesh-Agnostic
• Model Order Reduction
• Lyapunov Stability Regularization
• Latent Dynamics
• Time-Dependent PDEs