Spatio-temporal neural operator on complex geometries

In engineering and Artificial Intelligence (AI) scenarios, learning spatio-temporal dynamics is not only associated with high-resolution time series, but also often accompanied by complex computational domains. Consequently, learning high-dimensional spatio-temporal dynamics on complex geometries remains a significant challenge in both machine learning and engineering fields. Recently, Neural Operators have emerged as the lasted method which can learn mapping between functions with a discretisation resolution invariant network framework, and have increasingly been applied in engineering scenarios involving spatio-temporal dynamics. However, most existing neural operators require uniform grid data defined over regular spatio-temporal domains, making them inapplicable for engineering problems involving complex geometries. To address this limitation, we propose a Spatio-Temporal Neural Operator (STNO) that can learn mappings between functions defined simultaneously in the temporal domain and the complex geometric domain. The proposed STNO features a spatio-temporal iterative kernel integration module that separately encodes high-dimensional spatial information and temporal information into different low-dimensional frequency spaces, thus enabling efficient parameterised learning. Additionally, the model structure of STNO is independent of the discretisation resolution in both temporal and spatial domains. Experiments on several engineering case studies demonstrate the effectiveness and generality of the proposed STNO in learning spatio-temporal dynamics on complex geometries.