  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.
