Engineering design problems often involve solving parametric Partial Differential Equations (PDEs) under variable PDE parameters and domain geometry. Recently, neural operators have shown promise in learning PDE operators and quickly predicting the PDE solutions. However, training these neural operators typically requires large datasets, the acquisition of which can be prohibitively expensive. To overcome this, physics-informed training offers an alternative way of building neural operators, eliminating the high computational costs associated with Finite Element generation of training data. Nevertheless, current physics-informed neural operators struggle with limitations, either in handling varying domain geometries or varying PDE parameters. In this research, we introduce a novel method, the Physics-Informed Geometry-Aware Neural Operator (PI-GANO), designed to simultaneously generalize across both PDE parameters and domain geometries. We adopt a geometry encoder to capture the domain geometry features, and design a novel pipeline to integrate this component within the existing Deep Compositional Operator Network architecture. Numerical results demonstrate the accuracy and efficiency of the proposed method. All the codes and data related to this work are available on GitHub: https://github.com/uq-group/codes/tree/main/PI-GANO.

中文翻译：

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物理学感知几何感知神经运算符


工程设计问题通常涉及在可变 PDE 参数和域几何结构下求解参数化偏微分方程 （PDE）。最近，神经算子在学习 PDE 算子和快速预测 PDE 解方面显示出前景。然而，训练这些神经算子通常需要大型数据集，而获取这些数据集的成本可能高得令人望而却步。为了克服这个问题，物理信息训练提供了一种构建神经运算符的替代方法，消除了与有限元生成训练数据相关的高计算成本。然而，当前以物理学为依据的神经算子在处理不同的域几何结构或不同的 PDE 参数方面遇到了限制。在这项研究中，我们引入了一种新方法，即物理知情几何感知神经运算符 （PI-GANO），旨在同时泛化 PDE 参数和域几何。我们采用几何编码器来捕获域几何特征，并设计了一种新颖的管道，将此组件集成到现有的深度组合算子网络架构中。数值结果验证了所提方法的准确性和效率。与这项工作相关的所有代码和数据都可以在 GitHub 上找到：https://github.com/uq-group/codes/tree/main/PI-GANO。