Statistical learning of stochastic complex systems via the Yau-Yau nonlinear filter

Complex systems are characterized by nonlinearity, uncertainty, and noise interference, making it challenging to unravel their internal operation mechanisms. Here, we overcome this issue by implementing the Yau-Yau nonlinear filter theory to reconstruct stochastic networks from noisy data. Beyond ordinary differential equation-based deterministic networks, Yau-Yau stochastic networks can estimate the real states of actions and interactions among agents within dynamical systems and characterize how these states fluctuate due to stochastic perturbations. We perform Monte Carlo simulation to validate the statistical behavior of the Yau-Yau algorithm under different application scenarios. We employ the new approach to learn microbial stochastic interaction networks from a tri-cultural data of bacterial species, gaining new insight into how microbial interactions facilitate information processing and leverage robustness against perturbations. Yau-Yau networks open up a new avenue to reveal the emergence of order from disorder for complex systems.