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

2026-04-07

AI Summary

This work addresses the challenge of modeling nonlinear stochastic dynamical systems from noisy data by proposing a novel method that integrates deep symbolic regression with Gaussian process maximum likelihood estimation. Without requiring prior assumptions about the form of the dynamics or noise, the approach jointly infers the analytical structure of the governing equations and quantifies parametric uncertainty. It achieves, for the first time, simultaneous symbolic discovery of stochastic differential equations and identification of noise characteristics. Validated on benchmark systems—including the harmonic oscillator, Duffing, and van der Pol oscillators—as well as biological coupled-oscillator experiments, the method robustly recovers the true equations and noise structures using only 100–1000 data points, substantially enhancing accuracy and interpretability in small-sample stochastic system identification.

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.

Problem

• stochastic dynamics
• symbolic regression
• uncertainty quantification
• noisy data
• nonlinear systems

Innovation

• symbolic regression
• Gaussian processes
• stochastic dynamics
• uncertainty quantification
• machine learning