A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data

TL;DR

This paper introduces a hybrid machine learning framework combining symbolic regression and Gaussian processes to uncover stochastic nonlinear dynamics from noisy data, accurately identifying governing equations and noise characteristics.

Contribution

It develops a novel approach that simultaneously infers symbolic system equations and noise structure without prior assumptions, improving understanding of stochastic dynamical systems.

Findings

Successfully identified equations and noise in oscillators and biological systems.

Data-efficient, requiring only 100-1000 data points.

Robust to noise, demonstrating broad applicability.

Abstract

Modeling real-world systems requires accounting for noise - whether it arises from unpredictable fluctuations in financial markets, irregular rhythms in biological systems, or environmental variability in ecosystems. While the behavior of such systems can often be described by stochastic differential equations, a central challenge is understanding how noise influences the inference of system parameters and dynamics from data. Traditional symbolic regression methods can uncover governing equations but typically ignore uncertainty. Conversely, Gaussian processes provide principled uncertainty quantification but offer little insight into the underlying dynamics. In this work, we bridge this gap with a hybrid symbolic regression-probabilistic machine learning framework that recovers the symbolic form of the governing equations while simultaneously inferring uncertainty in the system parameters. The framework combines deep symbolic regression with Gaussian process-based maximum likelihood estimation to separately model the deterministic dynamics and the noise structure, without requiring prior assumptions about their functional forms. We verify the approach on numerical benchmarks, including harmonic, Duffing, and van der Pol oscillators, and validate it on an experimental system of coupled biological oscillators exhibiting synchronization, where the algorithm successfully identifies both the symbolic and stochastic components. The framework is data-efficient, requiring as few as 100-1000 data points, and robust to noise - demonstrating its broad potential in domains where uncertainty is intrinsic and both the structure and variability of dynamical systems must be understood.