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：

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物理知情离散化独立深度组合算子网络


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，已成功使用神经运算符，它预测具有可变 PDE 参数输入的 PDE 解。然而，神经算子的训练通常需要大型训练数据集，而获取这些数据集的成本可能高得令人望而却步。为了应对这一挑战，以物理为基础的培训可以提供一种具有成本效益的策略。然而，当前以物理学为依据的神经运算符面临局限性，无论是在处理不规则的域形状方面，还是在推广到 PDE 参数的各种离散表示方面。在这项研究中，我们引入了一种新的物理学模型架构，它可以推广到 PDE 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，我们的模型涉及对参数嵌入的重复离散化独立学习，并且这种参数嵌入通过多个组合层与响应嵌入集成，以获得更多的表现力。数值结果验证了所提方法的准确性和效率。