Title: Latent PDE Mapping for Shape-Generalizable Physics-Informed Neural Networks
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: physics-informed neural networks, cardiac electrophysiology, latent representation, variable geometry

Abstract: Physics-Informed Neural Networks (PINNs) have shown strong potential for learning physically consistent representations from sparse data, but often struggle to generalize to geometries with varying shapes. To address this challenge, we introduce $\textit{latent PDE mapping}$, a technique for mapping geometry-specific partial differential equations (PDEs) to a shared latent PDE representation using the deformation gradient. We embed latent PDE mapping into the PINN framework (LPM-PINN), enabling PINNs to capture geometric variability while preserving the governing physics. This integration facilitates accurate predictions of nonlinear, time-dependent systems even in geometries well beyond the training distribution. We demonstrate LPM-PINN on a challenging nonlinear time-dependent PDE with sharp gradients, the Aliev–Panfilov model of cardiac electrophysiology, in both 2D and 3D. Our results show that LPM-PINN generalizes robustly across diverse geometries, including shapes with drastically changing boundaries. These findings establish latent PDE mapping as a promising approach for extending PINNs to applications with variable geometries and complex physics.