Solving parametric Partial Differential Equations (PDEs) for a broad range of
parameters is a critical challenge in scientific computing. To this end, neural
operators, which learn mappings from parameters to solutions, have been
successfully used. However, the training of neural operators typically demands
large training datasets, the acquisition of which can be prohibitively
expensive. To address this challenge, physics-informed training can offer a
cost-effective strategy. However, current physics-informed neural operators
face limitations, either in handling irregular domain shapes or in
generalization to various discretizations of PDE parameters with variable mesh
sizes. In this research, we introduce a novel physics-informed model
architecture which can generalize to parameter discretizations of variable size
and irregular domain shapes. Particularly, inspired by deep operator neural
networks, our model involves a discretization-independent learning of parameter
embedding repeatedly, and this parameter embedding is integrated with the
response embeddings through multiple compositional layers, for more
expressivity. Numerical results demonstrate the accuracy and efficiency of the
proposed method.

参数化偏微分方程、神经算子、物理信息训练、不规则域形状和可变网格尺寸的研究