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

arXiv: https://arxiv.org/abs/2601.00834 · HTML: https://mlsanity.com/paper/2601.00834/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 55 references that are in the map.
- First posted on arXiv 26 Dec 2025.

## Builds on (most cited first, 28 of 28)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Deep Networks and the Multiple Manifold Problem](https://mlsanity.com/paper/2008.11245/) (2020)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Improving physics-informed neural networks with meta-learned optimization](https://mlsanity.com/paper/2303.07127/) (2023)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)
- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Predicting and Enhancing the Fairness of DNNs with the Curvature of Perceptual Manifolds](https://mlsanity.com/paper/2303.12307/) (2023)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)
- [Horizontal Flows and Manifold Stochastics in Geometric Deep Learning](https://mlsanity.com/paper/1909.06397/) (2019)
- [State Representation Learning Using an Unbalanced Atlas](https://mlsanity.com/paper/2305.10267/) (2023)
- [Deep Stochastic Mechanics](https://mlsanity.com/paper/2305.19685/) (2023)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)

## Abstract (arXiv)

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 ∈ [-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 Emass ≈ 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.