[2601.00834] Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds

Computer Science > Machine Learning arXiv:2601.00834 (cs) [Submitted on 26 Dec 2025 (v1), last revised 23 Mar 2026 (this version, v3)] Title:Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds Authors:Julian Evan Chrisnanto, Salsabila Rahma Alia, Nurfauzi Fadillah, Yulison Herry Chrisnanto View a PDF of the paper titled Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds, by Julian Evan Chrisnanto and 3 other authors View PDF HTML (experimental) Abstract:Simulating nonlinear reaction-diffusion dynamics on complex, non-Euclidean manifolds remains a fundamental challenge in computational morphogenesis, constrained by high-fidelity mesh generation costs and symplectic drift in discrete time-stepping schemes. This study introduces the Intrinsic-Metric Physics-Informed Neural Network (IM-PINN), a mesh-free geometric deep learning framework that solves partial differential equations directly in the continuous parametric domain. By embedding the Riemannian metric tensor into the automatic differentiation graph, our architecture analytically reconstructs the Laplace-Beltrami operator, decoupling solution complexity from geometric discretization. We validate the framework on a "Stochastic Cloth" manifold with extreme Gaussian curvature fluctuations ($K \in [-2489, 3580]$), where traditional adaptive refinement fails to resolve anisotropic T...