Physics-informed neural networks for transformed geometries and manifolds
This addresses a bottleneck in applying PINNs to complex geometries in science and engineering, though it is an incremental improvement over existing PINN methods.
The paper tackled the problem of physics-informed neural networks (PINNs) struggling with complex or varying geometries by proposing a method to integrate geometric transformations, which enhanced flexibility and allowed for shape optimization during training, as demonstrated on problems like the Eikonal equation on an Archimedean spiral and Stokes flow in a deformed tube.
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.