## IM-PINN: A New Approach to Geometric Deep Learning A new geometric deep learning framework, called Intrinsic-Metric Physics-Informed Neural Network (IM-PINN), has been presented, designed to dynamically simulate nonlinear reactions and diffusions on complex non-Euclidean manifolds. This system addresses the challenges associated with high-fidelity mesh generation and symplectic drift in traditional time-stepping schemes. ## Architecture and Functionality The IM-PINN uses a neural network that incorporates the Riemannian metric tensor into the automatic differentiation graph. This architecture analytically reconstructs the Laplace-Beltrami operator, decoupling the solution complexity from the geometric discretization. The system was validated on a "Stochastic Cloth" manifold with extreme Gaussian curvature fluctuations. ## Performance and Benefits Tests have shown that IM-PINN outperforms the Surface Finite Element Method (SFEM) in terms of physical rigor. IM-PINN achieves a global mass conservation error of approximately 0.157, compared to SFEM's 0.258. This makes it a thermodynamically consistent global solver, eliminating the 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.