Gaussian process regression with additive periodic kernels for two-body interaction analysis in coupled phase oscillators

Since many physical laws -- from classical mechanics to electromagnetism -- are formulated as two-body interactions, the same perspective naturally extends to biological and social dynamics. Here we focus on rhythmic phenomena, where phase reduction theory shows that synchronization dynamics can be universally described by coupled phase oscillators. Estimating the interaction functions of such systems from data offers a direct path to understanding and predicting such dynamics. Existing Fourier-series-based methods encounter difficulties with limited or biased data. To overcome this, we employ Gaussian process regression with additive periodic kernels. In our approach, we incorporate information about the estimation target into the statistical model in advance by designing kernel functions that capture the characteristics -- additivity and $2\pi$-periodicity -- of the coupling functions. Furthermore, owing to the Bayesian framework, our method enables the evaluation of uncertainty in the estimation results. We validate our approach on Van der Pol, FitzHugh-Nagumo, and spiking neural models. Our approach outperforms Fourier-series baselines in both error and stability under biased phase sampling. This enables data-driven studies of rhythm dynamics across a broader range of datasets. Furthermore, it makes a first step toward a statistically grounded, data-driven approach to general many-body systems with two-body interactions.