Data-driven reconstruction of spatiotemporal phase dynamics for traveling and oscillating patterns via Bayesian inference

Abstract: Building on the phase reduction theory formulated for reaction-diffusion systems with spatial translational symmetry, we develop a data-driven method that reconstructs the spatiotemporal phase dynamics of traveling and oscillating patterns. Spatiotemporal phase dynamics are described by spatial and temporal phases that represent the position and oscillation of the pattern, respectively. Using Bayesian inference, our method directly reconstructs phase equations from time-series data. When tested on simulation data from coupled Gray-Scott models exhibiting traveling breathers, the method accurately reconstructs the deterministic part of the phase equations in the weak-noise regime, in which the phase dynamics converge to a linearly stable fixed point.

• The paper introduces a data-driven technique that reconstructs reduced phase dynamics using Bayesian inference, eliminating the need for complete dynamical equations.
• It accurately extracts spatial and temporal phases via order parameter regression and linear interpolation, validated on Gray-Scott reaction-diffusion models.
• The method quantifies noise effects on phase coupling functions and offers broad applicability for analyzing synchronization in systems with propagating oscillatory patterns.

Data-Driven Reconstruction of Spatiotemporal Phase Dynamics in Traveling and Oscillating Patterns

Introduction

The paper "Data-driven reconstruction of spatiotemporal phase dynamics for traveling and oscillating patterns via Bayesian inference" (2604.23727) presents a methodology for reconstructing the reduced phase dynamics of traveling and oscillating patterns, specifically focusing on traveling breathers in reaction-diffusion systems. The principal innovation is a data-driven approach—rooted in phase reduction theory but not reliant on full knowledge of the underlying dynamical equations—that infers deterministic phase equations directly from observed time series. Core to this method is the joint reconstruction of spatial and temporal phase dynamics, exploiting the system's translational symmetry and oscillatory structure. Reconstruction is accomplished via a Bayesian inference scheme calibrated against numerical data from coupled Gray-Scott models.

Theoretical Framework: Spatiotemporal Phase Reduction

Phase reduction for reaction-diffusion PDEs with spatial translational symmetry yields ODEs for two interdependent phase variables: a spatial phase $\Phi(t)$ encoding position and a temporal phase $\Theta(t)$ encoding oscillatory activity. For traveling breathers, the underlying PDE is reduced to effective equations: $$\dot{\Phi}_i = c_i + \epsilon \sum_{j\neq i} \Gamma_{ij}^{\mathrm{s}}(\Phi_i-\Phi_j,\Theta_i-\Theta_j) + \eta_i^{\mathrm{s}}(t)$$

$$\dot{\Theta}_i = \omega_i + \epsilon \sum_{j\neq i} \Gamma_{ij}^{\mathrm{t}}(\Phi_i-\Phi_j,\Theta_i-\Theta_j) + \eta_i^{\mathrm{t}}(t)$$

where $\Gamma_{ij}^{\mathrm{s}}, \Gamma_{ij}^{\mathrm{t}}$ are phase coupling functions, and the $\eta$ terms capture projected noise. The coupling functions depend only on phase differences due to symmetry. These couplings can be represented as truncated double Fourier series.

Classic data-driven phase reduction approaches are inadequate for traveling, oscillating structures since they only consider oscillatory phase and neglect the crucial spatial phase. The authors systematically generalize data-driven reconstruction to multivariate phase variables, enabling the explicit analysis of collective behavior such as synchronization in spatially extended, moving patterns.

Extraction of Spatial and Temporal Phases

Accurate extraction of the spatial and temporal phases $\Phi_i(t)$ and $\Theta_i(t)$ from data is critical. The procedure is as follows:

• Temporal phase is estimated from an observable aggregate (akin to a spatial integral of a single field component) using linear interpolation of threshold crossing times (Poincaré section) rather than analytic signal approaches, which are shown to have inferior accuracy in the weak-noise regime.
• Spatial phase is computed via the argument of a suitably defined order parameter sensitive to translations, with regression to separate the phase dynamics into a "drift" (travel velocity) and a phase-dependent correction.
• The protocol accounts for the nonlinearity in phase-space time series, and harmonics are selected by decomposing both the spatial and temporal phase dependencies.

Bayesian Inference Algorithm for Phase Equation Reconstruction

Phase evolution data, $$\left\{\Phi_{i,n}^\ast, \Theta_{i,n}^\ast, \dot{\Phi}_{i,n}, \dot{\Theta}_{i,n}\right\}$$, is used with a Bayesian regression model, which fits the Fourier coefficients of phase coupling functions and estimates the noise covariance. The likelihood function incorporates the full structure necessitated by two coupled SDEs (correction for Jacobian, noise covariance). The prior distribution is Gaussian and uninformative, and inference is performed via iterative updates maximizing the posterior.

The method estimates not only the functional forms of the deterministic couplings but also the noise-induced drift shifts and correlations between phase noise components, essential for a correct macroscopic SDE description.

Numerical Tests: Weakly Coupled Gray-Scott Reaction-Diffusion Systems

The authors validate their methodology on simulation data from weakly coupled Gray-Scott reaction-diffusion models producing traveling breather solutions. The setup allows precise control of noise, parameters, and access to analytic ground truth phase reductions.

Key strong numerical results include:

• For noise intensity $\sigma^2 \leq 1.0 \times 10^{-10}$, phase coupling functions are reconstructed with high accuracy. Prediction errors for the couplings are much smaller than their magnitude.
• The reconstructed Fourier coefficients of the phase coupling functions match theoretical values, and irrelevant coefficients are negligibly small as expected.
• The method successfully reconstructs noise-induced constant shifts and the full noise covariance; estimated scaling with noise intensity matches theoretical predictions, although the absolute magnitude of estimated noise is systematically smaller.

In the strong noise regime ($\Theta(t)$0), accuracy is lost: reconstructed phase equations reflect the instability of phase locking, and prediction errors rise to the scale of the phase couplings themselves.

Implications and Future Prospects

Practically, the method offers a robust path to analyze synchronization and collective dynamics in spatiotemporal systems whose underlying equations are unavailable or only partially known—for example, climate dynamics involving propagating atmospheric waves, or experimental chemical patterns with moving spatial structure.

Key implications and prospects:

• Model Selection: Setting the truncation order of the Fourier decomposition (the harmonic cutoffs for spatial and temporal phase differences) should balance expressivity and regularization, with implications for avoiding overfitting in applications to high-dimensional data.
• Noise Modeling: Accurate modeling of inter-phase noise covariance is essential in quantitative stochastic phase reductions. This is both theoretically nontrivial and practically important; the results suggest that, under sufficiently weak and uncorrelated noise, one may simplify to diagonal covariance for efficiency.
• Generalization: The framework extends straightforwardly to more complex geometries (e.g., spherical or cylindrical), enabling analysis of laboratory or geophysical systems with propagating oscillatory structures; it is thus highly relevant for teleconnection analysis in climate science, traveling waves in neuroscience, or rotating convection in fluid mechanics.
• Alternative Phase Calculation: Linear interpolation methods for phase estimation are empirically superior in the weak-noise regime; signal-processing-based methods (e.g., Hilbert transform) unnecessarily increase error in extracting the phase dynamics most relevant for slow, weakly perturbed dynamics.

Methodologically, the work solidifies the interface between model-driven reduction and purely statistical data-driven inference, showing that accurate, interpretable phase reductions are possible in realistic, spatiotemporally extended systems provided symmetry and slowly evolving collective variables can be appropriately exploited.

Conclusion

This work establishes a concrete methodology for data-driven reconstruction of coupled spatial and temporal phase dynamics from time series generated by systems with propagating, oscillatory patterns. Capitalizing on phase reduction theory and Bayesian inference, the method provides highly accurate reconstructions in the weak-noise regime and yields detailed quantitative insight into both deterministic and stochastic aspects of spatiotemporal synchronization. This approach generalizes the reach of data-driven phase reduction methodology and is poised for significant impact in the quantitative study of complex, spatially structured dynamical phenomena where symmetry, oscillatory dynamics, and traveling structures are prominent features.