Natural and artificial complex systems—ranging from food webs to spin glasses to the spread of opinions—are composed of different elementary units. These units can interact with each other nonlinearly, which results in complex dynamics. One of the challenges in the modeling of such systems is the identification of not only pairwise but also, more importantly, higher-order interactions together with their strengths and directions from measured multivariate time series. In a significant leap forward, we present an innovative method that transforms our ability to analyze complex systems, enabling the detailed reconstruction of both pairwise and higher-order interactions.

Our method creates equations from multivariate time series data by employing Kramers-Moyal coefficients. Such coefficients are derived from a Taylor expansion of the master equation—a fundamental concept of statistical physics—that describes the evolution of the probability distribution of a stochastic process.

To demonstrate our approach and to showcase its versatility for field applications, we investigate time series derived from simulations of various high-dimensional model systems with preset higher-order interactions. These systems include a population growth model, a cancer growth model, and a phase oscillator network model. Our method accurately reconstructs interactions even if time series are contaminated with different types of noise mimicking observational errors.

This advancement not only deepens our understanding of the intricate dynamics within multidimensional systems but also enhances modeling and its predictive power. Our method opens new avenues for investigating the complex interplay of factors in various systems, heralding a new era of insight and predictive precision in the study of complex dynamical systems.