Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang
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Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems
Learning operators for parametric partial differential equations (PDEs) using
neural networks has gained significant attention in recent years. However,
standard approaches like Deep Operator Networks (DeepONets) require extensive
labeled data, and physics-informed DeepONets encounter training challenges. In
this paper, we introduce a novel physics-informed tailored finite point
operator network (PI-TFPONet) method to solve parametric interface problems
without the need for labeled data. Our method fully leverages the prior
physical information of the problem, eliminating the need to include the PDE
residual in the loss function, thereby avoiding training challenges. The
PI-TFPONet is specifically designed to address certain properties of the
problem, allowing us to naturally obtain an approximate solution that closely
matches the exact solution. Our method is theoretically proven to converge if
the local mesh size is sufficiently small and the training loss is minimized.
Notably, our approach is uniformly convergent for singularly perturbed
interface problems. Extensive numerical studies show that our unsupervised
PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised
deep operator networks in terms of accuracy and versatility.