Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains

Shizheng Wen (ETHZ - ETH Zurich) · Arsh Kumbhat · Levi Lingsch (ETH Zurich) · Sepehr Mousavi (ETHZ - ETH Zurich) · Yizhou Zhao (Carnegie Mellon University) · Praveen Chandrashekar (Tata Institute of Fundamental Research) · Siddhartha Mishra (Swiss Federal Institute of Technology)

accurate modelsarbitrary domainscomputational efficiencygeometry aware operator transformergeometry embeddingsindustrial simulationslearning tasksmultiscale attentional graph neural operatorsoperator learning algorithmspde solution approximationrobust approximationscalabilitystate of the art performancethree-dimensional cfd datasetsvision transformer processors

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on three large scale three-dimensional industrial CFD datasets. Our project page for accessing the source code is available at https://camlab-ethz.github.io/GAOT.