# High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation

arXiv: https://arxiv.org/abs/2601.08104 · HTML: https://mlsanity.com/paper/2601.08104/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 72 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [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)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Improving Reproducibility in Machine Learning Research (A Report from the NeurIPS 2019 Reproducibility Program)](https://mlsanity.com/paper/2003.12206/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations](https://mlsanity.com/paper/2203.11363/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Simulating Three-dimensional Turbulence with Physics-informed Neural Networks](https://mlsanity.com/paper/2507.08972/) (2025)
- [Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Bases](https://mlsanity.com/paper/2410.03496/) (2024)
- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [Spectral Informed Neural Network: An Efficient and Low-Memory PINN](https://mlsanity.com/paper/2408.16414/) (2024)
- [PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/2307.11289/) (2023)
- [A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem](https://mlsanity.com/paper/2309.04439/) (2023)
- [Physics-informed neural networks for transformed geometries and manifolds](https://mlsanity.com/paper/2311.15940/) (2023)
- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [Brain-Inspired Physics-Informed Neural Networks: Bare-Minimum Neural Architectures for PDE Solvers](https://mlsanity.com/paper/2401.15661/) (2024)
- [Feature Mapping in Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2402.06955/) (2024)
- [Impact of Loss Weight and Model Complexity on Physics-Informed Neural Networks for Computational Fluid Dynamics](https://mlsanity.com/paper/2509.21393/) (2025)

## Abstract (arXiv)

The accurate identification and control of spatiotemporal chaos in reaction-diffusion systems remains a grand challenge in chemical engineering, particularly when the underlying catalytic surface possesses complex, unknown topography. In the textitDefect Turbulence regime, system dynamics are governed by topological phase singularities (spiral waves) whose motion couples to manifold curvature via geometric pinning. Conventional Physics-Informed Neural Networks (PINNs) using ReLU or Tanh activations suffer from fundamental textitspectral bias, failing to resolve high-frequency gradients and causing amplitude collapse or phase drift. We propose a Multi-Scale SIREN-PINN architecture leveraging periodic sinusoidal activations with frequency-diverse initialization, embedding the appropriate inductive bias for wave-like physics directly into the network structure. This enables simultaneous resolution of macroscopic wave envelopes and microscopic defect cores. Validated on the complex Ginzburg-Landau equation evolving on latent Riemannian manifolds, our architecture achieves relative state prediction error εL2 ≈ 1.92 × 10-2, outperforming standard baselines by an order of magnitude while preserving topological invariants (|Delta Ndefects| < 1). We solve the ill-posed textitinverse pinning problem, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation ρ = 0.965). Training dynamics reveal a distinctive Spectral Phase Transition at epoch ~ 2,100, where cooperative minimization of physics and geometry losses drives the solver to Pareto-optimal solutions. This work establishes a new paradigm for Geometric Catalyst Design, offering a mesh-free, data-driven tool for identifying surface heterogeneity and engineering passive control strategies in turbulent chemical reactors.