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存在局限。我们提出一种新颖方法，通过将几何变换集成到PINNs中，以鲁棒地适应几何变化。该方法引入微分同胚作为参考域的映射，并调整物理信息损失函数的导数计算。这不仅将PINNs的适用性推广到光滑变形域与低维流形，还能在训练网络时直接进行形状优化。我们通过多个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动，以及（iv）基于拉普拉斯算子的形状优化。这些实例表明，该方法在几何变化场景下相较传统PINNs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，为科学工程中复杂几何体上的偏微分方程建模铺平了道路。