BV-NORM: A neural operator learning framework for parametric boundary value problems on complex geometric domains in engineering

Boundary Value Problems (BVPs) are extensively employed in engineering for process modelling and optimisation. These problems frequently involve complex geometries and require the massive solution of BVPs under different boundary conditions (BCs). Neural operators (NOs), capable of learning mappings between infinite-dimensional functions, present a potential solution for solving parametric BVPs. However, existing NOs are typically designed for scenarios where the input and output functions share the same domain, thus not applicable to BVPs in which the BC and solution functions are defined over different complex domains. Therefore, this study presents a novel deep learning framework called Boundary Value Neural Operator on Riemannian Manifolds (BV-NORM) for solving parametric BVPs involving complex geometric domains. BV-NORM introduces two sub-networks, Geometry-NET (Geo-NET) and Boundary condition-NET (BC-NET), to encode geometric information and boundary conditions. Consequently, the geometric information of the output domain can then be incorporated into the learning process. Furthermore, the Laplace kernel integration module of NORM is employed to construct the sub-networks, thereby enhancing the capacity to analyse complex geometric domains. The performance of the proposed method is evaluated in four benchmark cases, including toy Partial differential equation (PDE) cases and engineering applications, through comparisons with existing baseline neural operators. The experimental results validate that BV-NORM effectively addresses BVPs across various engineering scenarios.

Authors

• Zhiliang Deng
• Qinglu Meng
• Yingguang Li
• Xu Liu Watrix AI
• Gengxiang Chen
• Lu Chen Shanghai Jiao Tong University Shanghai Jiao Tong University (Wuxi) Blockchain Advanced Research Center
• Changqing Liu
• Xiaozhong Hao

Keywords

• Neural operators
• Boundary value problems
• Complex geometric domains
• Engineering applications