# Data-Driven Weak-form Discovery of Stochastic Systems

arXiv: https://arxiv.org/abs/2603.20904 · HTML: https://mlsanity.com/paper/2603.20904/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 14 references that are in the map.
- First posted on arXiv 21 Mar 2026.

## Builds on (most cited first, 4 of 4)

- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)

## Abstract (arXiv)

We present an algorithm for learning the governing equations of a stochastic dynamical system from trajectory data. It recovers interpretable symbolic expressions for both the drift b(x) and the diffusion a(x) in a single pass, yielding a model that can be queried directly for relaxation timescales, metastable escape rates, and stationary distributions. Rather than estimating the dynamics one time step at a time, the algorithm averages each candidate term across the whole trajectory before regressing; a drift-informed correction further removes the finite-sampling bias in the diffusion estimate, cutting it from 4.6% to 0.6% for state-dependent noise. We also show that the trajectory averaging must use a spatial rather than a temporal weighting: temporal weighting, as in existing weak-form methods, is biased for stochastic data with an error that grows with dataset size. On three benchmark systems -- the Ornstein--Uhlenbeck process, a double-well Langevin system, and a multiplicative-noise system -- the algorithm recovers all coefficients to within 5%, stationary densities to within 0.01 in total variation, and escape rates that match the true dynamics.