Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within PINNs to robustly accommodate geometric variations. Our method incorporates a diffeomorphism as a mapping of a reference domain and adapts the derivative computation of the physics-informed loss function. This generalizes the applicability of PINNs not only to smoothly deformed domains, but also to lower-dimensional manifolds and allows for direct shape optimization while training the network. We demonstrate the effectivity of our approach on several problems: (i) Eikonal equation on Archimedean spiral, (ii) Poisson problem on surface manifold, (iii) Incompressible Stokes flow in deformed tube, and (iv) Shape optimization with Laplace operator. Through these examples, we demonstrate the enhanced flexibility over traditional PINNs, especially under geometric variations. The proposed framework presents an outlook for training deep neural operators over parametrized geometries, paving the way for advanced modeling with PDEs on complex geometries in science and engineering.

翻译：物理信息神经网络（PINNs）能够有效将物理原理嵌入机器学习，但在处理复杂或交替变化的几何结构时往往面临挑战。我们提出了一种新颖方法，将几何变换集成到物理信息神经网络中，以稳健地适应几何变化。该方法引入微分同胚作为参考域映射，并相应调整物理信息损失函数的导数计算。这不仅将PINNs的适用性推广至平滑变形域，还拓展至低维流形，并能在训练网络时直接进行形状优化。我们通过以下若干问题验证了该方法的有效性：（i）阿基米德螺旋上的程函方程，（ii）曲面流形上的泊松问题，（iii）变形管道中的不可压缩斯托克斯流，以及（iv）拉普拉斯算子形状优化。通过这些示例，我们展示了该方法相比传统PINNs在几何变化条件下具有更强的灵活性。所提出的框架为在参数化几何结构上训练深度神经算子提供了新思路，为科学工程领域中复杂几何结构的偏微分方程高级建模铺平了道路。