Partial differential equations are often used in the spatial-temporal modeling of complex dynamical systems in many engineering applications. In this work, we build on the recent progress of operator learning and present a data-driven modeling framework that is continuous in both space and time. A key feature of the proposed model is the resolution-invariance with respect to both spatial and temporal discretizations. To improve the long-term performance of the calibrated model, we further propose a hybrid optimization scheme that leverages both gradient-based and derivative-free optimization methods and efficiently trains on both short-term time series and long-term statistics. We investigate the performance of the spatial-temporal continuous learning framework with three numerical examples, including the viscous Burgers' equation, the Navier-Stokes equations, and the Kuramoto-Sivashinsky equation. The results confirm the resolution-invariance of the proposed modeling framework and also demonstrate stable long-term simulations with only short-term time series data. In addition, we show that the proposed model can better predict long-term statistics via the hybrid optimization scheme with a combined use of short-term and long-term data.

翻译：偏微分方程常被用于复杂动力系统（如众多工程应用）的时空建模中。本研究基于算子学习领域的最新进展，提出一种在空间和时间上均具有连续性的数据驱动建模框架。该模型的关键特征在于对空间离散化与时间离散化均具有分辨率不变性。为提升校准模型的长期性能，我们进一步提出一种混合优化策略，该策略融合了基于梯度与无梯度优化方法，能够有效利用短期时间序列数据与长期统计数据进行训练。我们通过三个数值算例（包括黏性Burgers方程、Navier-Stokes方程和Kuramoto-Sivashinsky方程）验证了该时空连续学习框架的性能。实验结果证实了所提框架的分辨率不变性，并表明仅基于短期时间序列数据即可实现稳定的长期仿真。此外，通过结合使用短期与长期数据的混合优化策略，所提模型能够更准确地预测长期统计特征。