Learning spatiotemporal dynamics from sparse data via a high-order physics-encoded network

Abstract

Learning unknown or partially known dynamics has gained significant attention in scientific machine learning (SciML). This research is mainly driven by the inherent sparsity and noise in scientific data, which poses challenges to accurately modeling spatiotemporal systems. While recent physics-informed learning strategies have attempted to address this problem by incorporating physics knowledge as soft constraints, they often encounter optimization and scalability issues. To this end, we present a novel physics-encoded learning framework for capturing the intricate dynamical patterns of spatiotemporal systems from limited sensor measurements. Our approach centers on a deep convolutional-recurrent network, termed Π-block, which hard-encodes known physical laws (e.g., PDE structure and boundary conditions) into the learning architecture. Moreover, the high-order time marching scheme (e.g., Runge-Kutta fourth-order) is introduced to model the temporal evolution. We conduct comprehensive numerical experiments on a variety of complex systems to evaluate our proposed approach against baseline algorithms across two tasks: reconstructing high-fidelity data and identifying unknown system coefficients. We also assess the performance of our method under various noisy levels and using different finite difference kernels. The comparative results demonstrate the superiority, robustness, and stability of our framework in addressing these critical challenges in SciML.