AI 中文总结
SMORE提出一种网格无关的PDE降阶框架，利用李雅普诺夫稳定性正则化训练潜在动力学，实现稳定长时域预测，并在多个PDE问题上取得与先进基线相当的精度。
AI 中文摘要
时间相关偏微分方程（PDE）的高保真模拟计算成本高昂，这促使人们开发数据驱动的降阶代理模型，用于不确定性量化、设计优化、数据同化和最优控制等多查询任务。然而，现有的代理模型往往表现出较差的时域稳定性，这可能导致不稳定的滚动预测以及在反向传播过程中出现梯度爆炸，尤其是在多步长时域预测中。为解决这一问题，我们提出了SMORE，一个用于时间相关PDE模型降阶的网格无关框架。其潜在动力学通过李雅普诺夫引导的稳定性正则化进行训练，该正则化促进了稳定的长时域滚动预测。我们在所述结构假设下提供了理论保证。除了预测PDE演化之外，学习到的潜在动力学是可解释的且为线性或线性二次形式，这可能为数据同化和最优控制等下游任务带来益处。此外，我们的框架能够从初始条件的稀疏测量中预测连续的PDE解场。我们在一系列问题上评估了SMORE，包括波传播、纳维-斯托克斯方程和浅水方程。我们的结果表明，与包括DINo、FNO、CNO和Transolver在内的竞争基线相比，SMORE在相当的参数预算下提高了长时域滚动预测的泛化能力和经验鲁棒性，并实现了具有竞争力的精度。
英文摘要
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.
Comments52 pages
