# Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme

arXiv: https://arxiv.org/abs/2105.08449 · HTML: https://mlsanity.com/paper/2105.08449/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 18 May 2021.

## Builds on (most cited first, 7 of 7)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching](https://mlsanity.com/paper/1807.05748/) (2018)
- [Deep learning algorithm for data-driven simulation of noisy dynamical system](https://mlsanity.com/paper/1802.08323/) (2018)
- [Variational Deep Learning for the Identification and Reconstruction of Chaotic and Stochastic Dynamical Systems from Noisy and Partial Observations](https://mlsanity.com/paper/2009.02296/) (2020)

## Built on by (most cited first, 10 of 10)

- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [Anomaly Detection and Generation with Diffusion Models: A Survey](https://mlsanity.com/paper/2506.09368/) (2025)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems](https://mlsanity.com/paper/2508.12569/) (2025)
- [First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems](https://mlsanity.com/paper/2606.11138/) (2026)
- [An Adaptive Random Fourier Features approach Applied to Learning Stochastic Differential Equations](https://mlsanity.com/paper/2507.15442/) (2025)
- [Self-induced stochastic resonance: A physics-informed machine learning approach](https://mlsanity.com/paper/2510.22848/) (2025)
- [Learning Deterministic and Stochastic Forced Hamiltonian Systems](https://mlsanity.com/paper/2608.19688/) (2026)
- [One-Step Generative Surrogate Models via Block-Triangular Joint Drifting](https://mlsanity.com/paper/2609.26435/) (2026)

## Abstract (arXiv)

Stochastic differential equations (SDEs) are one of the most important representations of dynamical systems. They are notable for the ability to include a deterministic component of the system and a stochastic one to represent random unknown factors. However, this makes learning SDEs much more challenging than ordinary differential equations (ODEs). In this paper, we propose a data driven approach where parameters of the SDE are represented by a neural network with a built-in SDE integration scheme. The loss function is based on a maximum likelihood criterion, under order one Markov Gaussian assumptions. The algorithm is applied to the geometric brownian motion and a stochastic version of the Lorenz-63 model. The latter is particularly hard to handle due to the presence of a stochastic component that depends on the state. The algorithm performance is attested using different simulations results. Besides, comparisons are performed with the reference gradient matching method used for non linear drift estimation, and a neural networks-based method, that does not consider the stochastic term.