Latent PDE mapping for efficient physics-informed learning across geometries with limited data

In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data. Latent PDE mapping pulls back geometry-specific PDE residuals and boundary conditions to a predefined latent geometry via the deformation gradient, thereby enabling the automated calculation of geometry-consistent shape gradients that are missing in conventional physics-informed machine learning formulations. We demonstrate the utility of latent PDE mapping in solving the anisotropic Aliev-Panfilov PDE of cardiac electrophysiology using both physics-informed neural networks and physics-informed deep operator networks. The Aliev-Panfilov PDE serves as a challenging exemplar: a nonlinear, time-dependent PDE benchmark with sharp gradients that are expensive to capture using traditional numerical solvers. To represent the limited data regime, we train the networks using just fifteen geometric samples drawn from parameterized distributions in two and three spatial dimensions. While modest improvements appear for geometries parameterized by affine and shear deformations, latent PDE mapping demonstrates significant benefits on select geometric families, achieving a factor ~4-6 reduction in mean relative L2 error. Furthermore, our results show that the computational cost of applying latent PDE mapping was modest during network training, and negligible at inference. Taken together, our study highlights how latent PDE mapping facilitates the creation of generalizable physics-informed machine learning models from limited sets of training geometries.

HyperShape: Hyperelasticity Across Diverse Shapes

Hyperelastic deformations are highly sensitive to domain geometry and boundary conditions, making generalization across both a critical capability for neural operators applied to these problems. However, existing benchmarks for neural operators on hyperelasticity rely on simple or few geometries, which makes it difficult to assess this capability rigorously. To address this gap, we introduce HyperShape, an extensible framework designed to generate synthetic shapes and their corresponding hyperelastic simulation data, producing a suite of 2D and 3D datasets with adjustable complexity and controllable shape variations. This design enables systematic assessment of generalization across in-distribution, out-of-distribution, and synthetic-to-real transfer settings. Using this framework, we evaluated the performance of several state-of-the-art neural operators over diverse shape distributions. Our findings reveal that neural operators perform well on simple shapes but struggle as shape complexity, geometric diversity, and boundary condition variability increase, requiring large amounts of training data in such regimes. Performance degrades consistently and predictably with geometric complexity highlighting the need for further model development. As an open and extensible benchmark, HyperShape is designed to grow alongside the field: new geometries, material models, loading conditions, and evaluation settings can be easily incorporated to validate hyperelastic surrogate models.