Data-driven inference of coupling functions in oscillatory systems via physics-informed neural networks

Coupling of oscillators is a universal phenomenon observed across diverse fields, from power grids in engineering to neuronal oscillators and circadian rhythms in biology. Understanding their dynamics requires identifying how individual oscillators interact. Traditionally, inferring the coupling mechanism from observable time series data involves estimating the coupling function—describing the interaction mechanism—using preassigned basis functions such as trigonometric polynomials. However, this approach is limited by the lack of a definitive rule for selecting the optimal number of basis functions, relying instead on subjective user choices. Here, we found that no single optimal number of basis functions exists: While using more basis functions helps capture complex coupling mechanisms, it tends to overfit and degrade performance with sparse or noisy data. To overcome these limitations, we developed Inference of Coupling using physics-informed neural networks (IC-PINN). Unlike conventional approaches, IC-PINN can infer diverse coupling functions from sparse and noisy data without requiring manual tuning, making it broadly applicable to various real-world systems of coupled oscillators. IC-PINN successfully identified that the segmentation clock operates via a “Winner-Take-All” synchronization mechanism and accurately inferred the coupling function in the self-assembly of optically interacting nanorotors. By offering a powerful tool for studying coupled oscillators, IC-PINN can provide deeper insights into complex interacting systems across biology, physics, and beyond.