Phase autoencoder for rapid data-driven synchronization of rhythmic spatiotemporal patterns

We present a machine-learning method for data-driven synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems. Based on the phase autoencoder [Yawata {\it et al.}, Chaos {\bf 34}, 063111 (2024)], we map high-dimensional field variables of the reaction-diffusion system to low-dimensional latent variables characterizing the asymptotic phase and amplitudes of the field variables. This yields a reduced phase description of the limit cycle underlying the rhythmic spatiotemporal dynamics in a data-driven manner. We propose a method to drive the system along the tangential direction of the limit cycle, enabling phase control without inducing amplitude deviations. With examples of 1D oscillating spots and 2D spiral waves in the FitzHugh-Nagumo reaction-diffusion system, we show that the method achieves rapid synchronization in both reference-based and coupling-based settings. These results demonstrate the potential of data-driven phase description based on the phase autoencoder for synchronization of high-dimensional spatiotemporal dynamics.

💡 Research Summary

The paper introduces a data‑driven method for synchronizing rhythmic spatiotemporal patterns in reaction‑diffusion (RD) systems by extending the recently proposed “phase autoencoder” (Yawata et al., Chaos 34, 063111 2024). The authors first review the classical phase‑amplitude reduction theory, which requires knowledge of the asymptotic phase function and its gradient (the phase response curve) and is therefore limited to systems with known governing equations. To overcome this limitation, they design a physics‑informed autoencoder that maps high‑dimensional field data (X(x,t)) to a low‑dimensional latent vector ((\theta, r_1,\dots,r_{D_L-1})). The first two latent components encode the asymptotic phase (\theta) via (\theta = \arctan(Y_1/Y_0)); the remaining components represent decaying amplitude modes with exponential rates (\lambda_j). During training, the natural frequency (\omega) and the Floquet exponents (\lambda_j) are simultaneously learned together with the encoder‑decoder weights.

Three key engineering improvements enable the method to handle large‑scale spatiotemporal data. First, convolutional layers replace fully‑connected layers, allowing the network to capture spatial correlations efficiently and drastically reducing the number of trainable parameters. Second, the latent dimension (D_L) is made flexible and can be increased far beyond the original three‑dimensional setting, providing sufficient capacity to represent multiple amplitude modes. Third, Gaussian noise is added to the inputs during training, encouraging nearby states in the original space to be mapped to nearby points in latent space, which stabilizes phase estimation against measurement noise.

After the autoencoder is trained, the authors propose a “tangential driving” control scheme. The external input is applied only in the direction of the limit‑cycle tangent (\chi’(\theta)), i.e., the derivative of the reconstructed pattern with respect to phase. Under this restriction the phase dynamics reduce to (\dot\theta = \omega + p(t)) while all amplitude dynamics follow (\dot r_j = \lambda_j r_j) and thus decay naturally. Consequently, the control law does not require explicit knowledge of the phase response curve or the isostable response functions, which are notoriously difficult to estimate in high‑dimensional settings.

The methodology is validated on the FitzHugh‑Nagumo RD model. Two benchmark patterns are considered: (i) a one‑dimensional oscillating spot and (ii) a two‑dimensional spiral wave. For each pattern the authors test two synchronization scenarios. In the reference‑based case a target phase trajectory is pre‑computed and the system is driven toward it; in the coupling‑based case two identical systems are weakly coupled and the control aims to minimize the phase difference. In both scenarios the tangential driving achieves synchronization orders of magnitude faster than conventional phase‑locking techniques, while the amplitudes remain essentially unchanged, demonstrating the robustness of the approach.

The paper’s contributions are threefold: (1) an extended phase autoencoder architecture capable of handling high‑dimensional spatiotemporal data, (2) a control strategy that exploits the learned phase representation without requiring phase‑sensitivity gradients, and (3) empirical evidence of rapid, amplitude‑preserving synchronization in both 1‑D and 2‑D RD systems. Limitations include the need for sufficiently rich, periodic training data, potential vulnerability to strong disturbances or non‑periodic transients where tangential driving may lose stability, and the fact that the method has so far been demonstrated only on relatively simple RD models. Future work should explore (i) incorporation of nonlinear amplitude‑phase coupling, (ii) real‑time implementation on experimental chemical or cardiac media, and (iii) robustness enhancements for noisy or partially observed data.