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

TL;DR

This paper introduces a multi-scale SIREN-PINN framework that accurately models stochastic chemical dynamics on complex manifolds, effectively capturing high-frequency wave features and reconstructing hidden curvature fields from limited data.

Contribution

The work develops a novel multi-scale SIREN-PINN architecture with sinusoidal activations for high-fidelity modeling of wave-like physics on complex manifolds, outperforming standard methods and solving inverse problems.

Findings

Achieves low relative prediction error (~1.92%) on complex Ginzburg-Landau equations.

Successfully reconstructs Gaussian curvature fields from partial wave observations.

Identifies a spectral phase transition during training that guides the optimization process.

Abstract

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 \textit{Defect 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 \textit{spectral 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 $\epsilon_{L_2} \approx 1.92 \times 10^{-2}$, outperforming standard baselines by an order of magnitude while preserving topological invariants ($|\Delta N_{defects}| < 1$). We solve the ill-posed \textit{inverse pinning problem}, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation $\rho = 0.965$). Training dynamics reveal a distinctive Spectral Phase Transition at epoch $\sim 2,100$, 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.