Latent PDE Mapping Boosts Physics-Informed AI for Varied Geometries

This study presents "latent PDE mapping," a novel physics-informed learning technique designed to enhance the efficiency of AI models in generalizing across different geometries, especially when training data is limited. The core idea involves transforming geometry-specific partial differential equation (PDE) residuals and boundary conditions back to a predefined latent geometry using the deformation gradient. This transformation allows for the automated calculation of geometry-consistent shape gradients, a feature often missing in conventional physics-informed machine learning. The utility of this method was demonstrated by solving the anisotropic Aliev-Panfilov PDE, a challenging, nonlinear, time-dependent equation relevant to cardiac electrophysiology, using both physics-informed neural networks and deep operator networks. Even with a sparse dataset of only fifteen geometric samples, latent PDE mapping achieved a significant reduction in mean relative L2 error (approximately 4-6 times) on certain geometric families. The computational overhead during training was modest, and negligible during inference, highlighting its practical efficiency for creating generalizable physics-informed models from limited training geometries.