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 DCON 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/WeihengZ/Physics-informed-Neural-Foundation-Operator.

翻译：工程设计问题通常涉及在变化的偏微分方程（PDE）参数和计算域几何形状下求解参数化偏微分方程。近年来，神经算子在学习和快速预测PDE解方面展现出潜力。然而，训练这些神经算子通常需要大规模数据集，而获取这些数据的成本可能极高。为克服此限制，物理信息训练提供了一种构建神经算子的替代方法，避免了通过有限元方法生成训练数据所带来的高昂计算成本。然而，当前的物理信息神经算子存在局限，难以同时处理变化的计算域几何形状和变化的PDE参数。在本研究中，我们提出了一种新方法——物理信息几何感知神经算子（PI-GANO），旨在同时对PDE参数和计算域几何形状实现泛化。我们采用几何编码器来捕捉计算域几何特征，并设计了一种新颖的流程，将该组件集成到现有的DCON架构中。数值结果验证了所提方法的准确性与效率。本工作相关的所有代码和数据均可在GitHub上获取：https://github.com/WeihengZ/Physics-informed-Neural-Foundation-Operator。