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 predicts the PDE solution with variable PDE parameter inputs, 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 in generalizing to various discrete representations of PDE parameters. In this research, we introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters 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.

中文翻译：

---

物理信息离散化独立的深度组合算子网络


求解各种参数的参数偏微分方程 (PDE) 是科学计算中的一项关键挑战。为此，神经算子已被成功使用，它可以通过可变 PDE 参数输入来预测 PDE 解。然而，神经算子的训练通常需要大量的训练数据集，而获取这些数据集的成本可能非常昂贵。为了应对这一挑战，物理知识培训可以提供一种经济有效的策略。然而，当前的物理信息神经算子在处理不规则域形状或推广到偏微分方程参数的各种离散表示方面都面临局限性。在这项研究中，我们引入了一种新颖的物理信息模型架构，它可以推广到偏微分方程参数和不规则域形状的各种离散表示。特别是，受深度算子神经网络的启发，我们的模型涉及到参数嵌入的离散化独立学习，并且该参数嵌入通过多个组合层与响应嵌入集成，以获得更多表现力。数值结果证明了该方法的准确性和效率。