FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition

Partial differential equations (PDEs) are integral in modeling and describing the dynamics of many complex systems in science and engineering. Numerical solvers such as finite element methods (FEMs) and finite difference methods (FDMs) often obtain the solution of PDEs by discretizing the domain and solving a finite-dimensional problem. However, obtaining high-resolution solutions to PDEs using numerical simulations for complex large-scale problems can be computationally expensive and prohibitive. There has been a growing interest in more efficient data-driven alternatives that can directly learn the underlying solutions from the available data without requiring explicit knowledge about the governing PDEs [3, 11]. More recently, operator learning has emerged as a promising paradigm, aiming to learn an unknown mathematical operator governing a system of PDEs [4]. They capture mappings between infinite-dimensional function spaces and have demonstrated potential in capturing complex solution behaviors [18, 15]. Furthermore, due to their inherent differentiability, they can be seamlessly applied to inverse problems, such as design optimization tasks [1]. Various architectures have been developed, including the Deep Neural Operator (DeepONet) [18], Fourier Neural Operator (FNO) [15], Graph Neural Operator [16], General Neural Operator Transformer (GNOT) [9] and Operator Transformer (OFormer) [14]. These models differ in their discretization methods and the approximation techniques they use to enhance efficiency and scalability.