Diffeomorphism Neural Operator for various domains and parameters of partial differential equations

Many science and engineering applications demand partial differential equations (PDE) evaluations that are traditionally computed with resource-intensive numerical solvers. Neural operator models provide an efficient alternative by learning the governing physical laws directly from data in a class of PDEs with different parameters, but constrained in a fixed boundary (domain). Many applications, such as design and manufacturing, would benefit from neural operators with flexible domains when studied at scale. Here we present a diffeomorphism neural operator learning framework towards developing domain-flexible models for physical systems with various and complex domains. Specifically, a neural operator trained in a shared domain mapped from various domains of fields by diffeomorphism is proposed, which transformed the problem of learning function mappings in varying domains (spaces) into the problem of learning operators on a shared diffeomorphic domain. Meanwhile, an index is provided to evaluate the generalization of diffeomorphism neural operators in different domains by the domain diffeomorphism similarity. Experiments on statics scenarios (Darcy flow, mechanics) and dynamic scenarios (pipe flow, airfoil flow) demonstrate the advantages of our approach for neural operator learning under various domains, where harmonic and volume parameterization are used as the diffeomorphism for 2D and 3D domains. Our diffeomorphism neural operator approach enables strong learning capability and robust generalization across varying domains and parameters.

Residual fourier neural operator for thermochemical curing of composites

During the curing process of composites, the temperature history heavily determines the evolutions of the field of degree of cure as well as the residual stress, which will further influence the mechanical properties of composite, thus it is important to simulate the real temperature history to optimize the curing process of composites. Since thermochemical analysis using Finite Element (FE) simulations requires heavy computational loads and data-driven approaches suffer from the complexity of highdimensional mapping. This paper proposes a Residual Fourier Neural Operator (ResFNO) to establish the direct high-dimensional mapping from any given cure cycle to the corresponding temperature histories. By integrating domain knowledge into a time-resolution independent parameterized neural network, the mapping between cure cycles to temperature histories can be learned using limited number of labelled data. Besides, a novel Fourier residual mapping is designed based on mode decomposition to accelerate the training and boost the performance significantly. Several cases are carried out to evaluate the superior performance and generalizability of the proposed method comprehensively.