SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs
cs.LG, cs.AI
Submitted: 2026-09-27
Updated: 2026-09-27
Comments: 52 pages
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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
Terminology
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.
Sources
- Transformer for Partial Differential Equations' Operator Learning
- CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- Fourier Neural Operator for Parametric Partial Differential Equations
- Learning Mesh-Based Simulation with Graph Networks
- CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations
- Continuous PDE Dynamics Forecasting with Implicit Neural Representations
- CoNFiLD: Conditional Neural Field Latent Diffusion Model Generating Spatiotemporal Turbulence
- A Survey on Diffusion Models for Time Series and Spatio-Temporal Data
- Modern Koopman Theory for Dynamical Systems
- LoRA: Low-Rank Adaptation of Large Language Models
- Compressed Convolutional LSTM: An Efficient Deep Learning framework to Model High Fidelity 3D Turbulence
- Operator is the Model
- Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows
- Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference
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