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Learning neural operators on Riemannian manifolds
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Learning neural operators on Riemannian manifolds
Learning mappings between functions(operators)defined on complex computational domains is a common theo-retical challenge in machine learning.Existing operator learning methods mainly focus on regular computational domains,and have many components that rely on Euclidean structural data.However,many real-life operator learning problems involve complex computational domains such as surfaces and solids,which are non-Euclidean and widely referred to as Riemannian manifolds.Here,we report a new concept,neural operator on Riemannian manifolds(NORM),which generalises neural op-erator from Euclidean spaces to Riemannian manifolds,and can learn the operators defined on complex geometries while pre-serving the discretisation-independent model structure.NORM shifts the function-to-function mapping to finite-dimensional mapping in the Laplacian eigenfunctions'subspace of geometry,and holds universal approximation property even with only one fundamental block.The theoretical and experimental analyses prove the significant performance of NORM in operator learning and show its potential for many scientific discoveries and engineering applications.
deep learningneural operatorpartial differential equationsRiemannian manifold
Gengxiang Chen、Xu Liu、Qinglu Meng、Lu Chen、Changqing Liu、Yingguang Li
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College of Mechanical and Electrical Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
School of Mechanical and Power Engineering,Nanjing Tech University,Nanjing 211816,China
deep learning neural operator partial differential equations Riemannian manifold
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