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

TL;DR

This paper introduces IM-PINN, a mesh-free neural network framework that accurately simulates reaction-diffusion dynamics on complex Riemannian manifolds, overcoming traditional discretization challenges and ensuring physical consistency.

Contribution

The study presents a novel intrinsic-metric physics-informed neural network that directly solves PDEs on manifolds by embedding the Riemannian metric, enabling mesh-free, high-fidelity simulations of complex geometries.

Findings

IM-PINN accurately recovers Turing patterns on complex manifolds.

The framework achieves lower mass conservation error than SFEM.

IM-PINN acts as a thermodynamically consistent global solver.

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 Turing instabilities. Using a dual-stream architecture with Fourier feature embeddings to mitigate spectral bias, the IM-PINN recovers the "splitting spot" and "labyrinthine" regimes of the Gray-Scott model. Benchmarking against the Surface Finite Element Method (SFEM) reveals superior physical rigor: the IM-PINN achieves global mass conservation error of $\mathcal{E}_{mass} \approx 0.157$ versus SFEM's $0.258$, acting as a thermodynamically consistent global solver that eliminates mass drift inherent in semi-implicit integration. The framework offers a memory-efficient, resolution-independent paradigm for simulating biological pattern formation on evolving surfaces, bridging differential geometry and physics-informed machine learning.