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

The paper tackles spatiotemporal chaos of the Curvature-Perturbed Stochastic Complex Ginzburg-Landau equation on rough manifolds by introducing a Multi-Scale SIREN-PINN that embeds a spectral sine-based inductive bias and a dual-branch geometry-state network. It demonstrates high-fidelity forward simulation and a robust inverse problem: reconstructing latent curvature from chaotic wave data (inverse pinning), revealing Curvature-Induced Pattern Selection as the geometry-driven regulator of defect turbulence. The framework achieves order-of-magnitude improvements in relative $L_2$ error over baselines, preserves topological invariants (defect counts), and recovers Gaussian curvature with $ ho=0.965$, while exposing a distinct Spectral Phase Transition during training that couples physics and geometry losses. These results establish a new paradigm for Geometric Catalyst Design and differentiable digital twins in turbulent chemical reactors and potentially in cardiac dynamics, where surface curvature governs pattern formation and control strategies.

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 $ε_{L_2} \approx 1.92 \times 10^{-2}$, outperforming standard baselines by an order of magnitude while preserving topological invariants ($|ΔN_{defects}| < 1$). We solve the ill-posed \textit{inverse pinning problem}, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation $ρ= 0.965$). 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.

Figures (9)

• Figure 1: Ground Truth Spatiotemporal Evolution. (a) GT Curvature showing the spatial variation in curvature across the domain. (b) GT Magnitude displaying the amplitude field distribution with predominantly low values throughout most of the domain. (c) GT Phase demonstrating the phase field structure with characteristic spatial patterns.
• Figure 2: Schematic of the Multi-Scale Dual-Stream SIREN-PINN Architecture. The model features two parallel pathways: a Geometry Branch ($\mathcal{N}_\kappa$) initialized with high-frequency Fourier modes to resolve surface roughness, and a State Branch ($\mathcal{N}_A$) for phase field evolution. The branches fuse within the automatic differentiation graph to compute the physics-informed residual $\mathcal{F}$, enabling simultaneous minimization of the data loss $\mathcal{L}_{Data}$ and the spectral PDE loss $\mathcal{L}_{PDE}$.
• Figure 3: Inverse Geometry Discovery and Forward Prediction. The SIREN-PINN architecture demonstrates its dual capability of simultaneously solving the inverse and forward problems in geometrically-modulated chaotic systems. Panel (a) shows the ground truth curvature perturbation $\kappa_{GT}(\mathbf{x})$ generated via Gaussian Random Fields, representing the hidden topographical landscape that modulates the reaction-diffusion dynamics. Panel (b) displays the latent curvature field $\kappa_\phi(\mathbf{x})$ recovered by the Geometry Branch, which was learned purely from observing the spatiotemporal patterns without direct access to the underlying manifold structure. The remarkable structural similarity ($\rho=0.965$, $L_2$ error = $4.3 \times 10^{-2}$) confirms that the network has successfully inverted the geometric pinning mechanism, identifying how persistent spiral cores betray the presence of concave basins while rapid phase scattering reveals convex peaks. Panel (c) presents the pointwise reconstruction error map, which reveals an intriguing physical insight: discrepancies concentrate primarily in quasi-flat regions where $|\kappa| \approx 0$. This spatially heterogeneous error distribution is not a failure but rather evidence of physically-informed learning—the model naturally prioritizes accuracy in high-curvature regions where the geometric coupling term $\kappa \cdot \nabla^2 A$ most strongly influences the wave dynamics, while exhibiting greater uncertainty in flat regions where the vanishing geometry gradient $\nabla_\kappa \mathcal{L}_{PDE} \to 0$ provides weak supervisory signal. Panel (d) demonstrates the forward prediction capability by showing a representative snapshot of the phase field $\phi(\mathbf{x},t)$ generated by the trained SIREN-PINN, illustrating how the recovered geometry field enables accurate long-term forecasting of the chaotic spiral turbulence patterns. The success of this blind reconstruction validates the effectiveness of the Total Variation regularization in disentangling deterministic geometric forcing from stochastic noise, essentially performing unsupervised source separation of the two competing instability mechanisms that drive pattern formation.
• Figure 4: Training Dynamics of the Inverse Solver. (a) The total loss trajectory reveals a "plateau-drop" behavior, staying flat for $\sim$2,000 epochs before a sudden phase transition drives the error down by four orders of magnitude. (b) The component-wise breakdown shows that the Dynamics Loss (Blue) and Curvature Loss (Red) are tightly coupled; the geometry cannot be learned until the physics residuals collapse, and the physics cannot be resolved without the correct geometry, leading to a simultaneous "cooperative" convergence event at Epoch 2,100.
• Figure 5: Failure Modes of Baseline Geometry Estimators. (a) The ReLU-PINN produces a "shattered" curvature field dominated by high-frequency noise due to its inability to model higher-order derivatives. (b) The Tanh-PINN exhibits similar "spectral pollution," failing to converge to a smooth manifold. (c) The Fourier-PINN generates a smoother field but suffers from amplitude over-estimation (Gibbs ringing), predicting curvature values double the magnitude of the ground truth. These artifacts render the baselines useless for identifying the true physical mechanism of defect pinning.