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 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 spectral bias, failing to resolve high-frequency gradients and causing amplitude collapse or phase drift. We propose a multiscale 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\epsilon _{L_2} \approx 1.92 \times 10^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϵ</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </msub> <mo>≈</mo> <mn>1.92</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, outperforming standard baselines by an order of magnitude while preserving topological invariants (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$|\Delta N_{defects}| < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> </mrow> <msub> <mi>N</mi> <mrow> <mi mathvariant="italic">defects</mi> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>). We solve the ill-posed inverse pinning problem, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho = 0.965$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.965</mn> </mrow> </math></EquationSource> </InlineEquation>). Training dynamics reveal a distinctive spectral phase transition at epoch <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sim 2,100$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>2</mn> <mo>,</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation>, 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.

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

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发表日期：

2026年7月6日

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Julian Evan Chrisnanto,
Salsabila Rahma Alia,
Nurfauzi Fadillah,
Yulison Herry Chrisnanto

摘要

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 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 spectral bias, failing to resolve high-frequency gradients and causing amplitude collapse or phase drift. We propose a multiscale 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 $\epsilon _{L_2} \approx 1.92 \times 10^{-2}$ $ϵ_{L_{2}} \approx 1.92 \times 10^{- 2}$ , outperforming standard baselines by an order of magnitude while preserving topological invariants ( $|\Delta N_{defects}| < 1$ $| Δ N_{defects} | < 1$ ). We solve the ill-posed inverse pinning problem, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation $\rho = 0.965$ $ρ = 0.965$ ). Training dynamics reveal a distinctive spectral phase transition at epoch $\sim 2,100$ $\sim 2, 100$ , 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.