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

TL;DR

This work tackles the challenge of simulating nonlinear reaction-diffusion dynamics on complex, non-Euclidean manifolds by introducing the Intrinsic-Metric Physics-Informed Neural Network (IM-PINN), a mesh-free solver that encodes the Riemannian metric into the automatic differentiation graph to reconstruct the Laplace-Beltrami operator $\Delta_{\mathcal{M}}$. The method employs a dual-stream architecture with Fourier feature embeddings to overcome spectral bias and enforces global mass conservation via a dedicated loss term $\mathcal{L}_{Mass}$, achieving thermodynamically consistent simulations that outperform a Surface Finite Element Method baseline in mass conservation. Validated on a highly curved Stochastic Cloth with $K$ ranging from $-2489$ to $3580$, IM-PINN reproduces anisotropic Turing patterns (e.g., splitting spots and labyrinthine regimes) tied to intrinsic geometry, while maintaining a fixed memory footprint and enabling resolution-independence. The results demonstrate a crucial shift from mesh-dependent discretization to geometry-aware, differentiable solvers, with implications for inverse design, surrogate modeling, and morphogenesis on evolving surfaces.

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.

Figures (7)

• Figure 1: The Stochastic Cloth Manifold ($\mathcal{M}$). Visualization of the computational domain generated by the parametric height function $z(u,v)$, illustrating the multi-scale geometric complexity arising from the superposition of deterministic wrinkles ($\omega_k$) and stochastic Gaussian perturbations ($\mathcal{G}$). The color map represents the surface elevation $z$, highlighting the non-trivial curvature landscape that the IM-PINN must resolve without a mesh.
• Figure 2: Spatial Modulation of the Chemical Potential. (a) The scalar field $\phi(u,v)$ defined in the parametric domain $\Omega=[0,1]^2$, showing the intrinsic periodic structure before geometric deformation. (b) The same field mapped onto the stochastic cloth manifold $\mathcal{M}$. The color intensity represents the magnitude of the potential, which locally modulates the feed rate $F$ of the Gray-Scott system. This visualization confirms that the IM-PINN must learn to resolve pattern variations that are driven by both the extrinsic curvature of the surface (wrinkles) and this intrinsic chemical heterogeneity.
• Figure 3: Schematic of the Intrinsic-Metric PINN (IM-PINN) Architecture. The diagram illustrates the dual-stream information flow. The Neural Stream (top) embeds coordinates into Fourier features to approximate the solution fields $\hat{U}, \hat{V}$. The Geometric Stream (bottom, dashed) explicitly encodes the Riemannian structure of the Stochastic Cloth, feeding the metric tensor $g_{ij}$ into the Automatic Differentiation (AD) engine. This enables the exact computation of the Laplace-Beltrami operator $\Delta_{\mathcal{M}}$ used in the composite loss function, which also incorporates the spatially varying chemical potential $\phi$ and mass conservation constraints.
• Figure 4: Intrinsic Phase Separation in the Parametric Domain $\Omega$. The contour plots display the learned concentrations of (a) the inhibitor $U$ and (b) the activator $V$ in the intrinsic $(u,v)$ coordinates. This view clearly shows the "Splitting Spot" regime of the Gray-Scott model. The periodic variation in pattern density (alternating between dense stripes and sparse spots) correlates directly with the underlying chemical potential field $\phi(u,v)$, confirming the IM-PINN's sensitivity to spatially modulated physics.
• Figure 5: Training Dynamics of the IM-PINN. (a) The total loss shows rapid initial convergence followed by fine-tuning. (b) The PDE residual loss drops by seven orders of magnitude, confirming that the network satisfies the local governing equations. (d) The component breakdown reveals that the boundary conditions ($\mathcal{L}_{BC}$, green) are satisfied almost to machine precision ($\sim 10^{-8}$), while the Mass Loss ($\mathcal{L}_{Mass}$, red) acts as a persistent regularizer, preventing the solution from drifting into unphysical regimes.