Learning stochasticity via a nonparametric approach to state-dependent noise estimation.

Modeling stochastic dynamical systems remains a central challenge across disciplines. Incomplete knowledge of nonlinear interactions and state-dependent fluctuations often renders bottom-up approaches ineffective, motivating methods that infer governing equations directly from data. However, parametric models struggle in the absence of strong prior assumptions, particularly when the intensity of the process noise depends on the system state. Here we introduce Trine (Three-phase Regression for INferred noisE), a nonparametric, kernel-based framework for inferring state-dependent noise from time-series data. The approach naturally extends to supervised learning settings with heteroskedastic noise. Trine employs a three-stage algorithm combining analytically solvable subproblems with a structured kernel architecture capable of capturing both abrupt stochastic fluctuations and smooth variations in variance. We validate Trine on biological and ecological systems, where it uncovers hidden dynamics without predefined parametric assumptions. Across benchmark systems, Trine achieves near-oracle performance, matching an idealized observer with direct access to the stochastic input realizations. On live-cell RNA transcription trajectories, Trine reveals localized regimes of elevated stochastic activity, demonstrating how effective state-dependent diffusion models can uncover biologically meaningful hidden variability directly from experimental observations. The Trine framework thus opens new avenues for quantifying how process noise shapes the behavior of complex dynamical systems.