Riesz Neural Operator for Solving Partial Differential Equations

Neural operators (NOs) have emerged as a new paradigm for efficiently solving partial differential equations (PDEs) in various scientific and engineering disciplines. However, the training of NOs relies on large numbers of highfidelity data generated by conventional numerical methods, which restricts the applications of NOs in complex physical systems due to prohibitive computational cost of data generation. In this study, we propose a self-supervised neural operator (SNO) framework aimed at alleviating this limitation by enabling the generation of training data without repeated use of numerical solvers. The SNO consists of three submodels: the first is a physics-informed sampler (PI-sampler) based on the Bayesian physics-informed neural networks (B-PINNs), which enables efficient and solver-free data generation, the second is the function encoder (FE) that learns compact representations for the inputs as well as outputs in learning operators, and the last submodel is an encoder-only Transformer for operator learning, which learns the mapping from different boundary/initial conditions, source term, and/or geometries to the solution of a specific PDE. We demonstrate the effectiveness of SNO using examples of one-dimensional nonlinear reaction-diffusion equations, two-dimensional Poisson equation on parameterized geometries, as well as two-and five-dimensional time-dependent PDEs. We also apply the SNO to a vortex-induced vibration of a flexible cylinder, which is one of the canonical problems in fluid dynamics and ocean engineering. The SNO