# Deep learning algorithm for data-driven simulation of noisy dynamical system

arXiv: https://arxiv.org/abs/1802.08323 · HTML: https://mlsanity.com/paper/1802.08323/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2018.

## Built on by (most cited first, 12 of 12)

- [LLM4ED: Large Language Models for Automatic Equation Discovery](https://mlsanity.com/paper/2405.07761/) (2024)
- [Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme](https://mlsanity.com/paper/2105.08449/) (2021)
- [An energy-based deep splitting method for the nonlinear filtering problem](https://mlsanity.com/paper/2203.17153/) (2022)
- [MetaPhysiCa: OOD Robustness in Physics-informed Machine Learning](https://mlsanity.com/paper/2303.03181/) (2023)
- [Incorporating Transformer and LSTM to Kalman Filter with EM algorithm for state estimation](https://mlsanity.com/paper/2105.00250/) (2021)
- [Discovering Physics Laws of Dynamical Systems via Invariant Function Learning](https://mlsanity.com/paper/2502.04495/) (2025)
- [SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics](https://mlsanity.com/paper/2302.03663/) (2023)
- [Generative Adversarial Network for Probabilistic Forecast of Random Dynamical System](https://mlsanity.com/paper/2111.03126/) (2021)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [Deep Chaos Synchronization](https://mlsanity.com/paper/2104.08436/) (2021)
- [Deep Learning for Stability Analysis of a Freely Vibrating Sphere at Moderate Reynolds Number](https://mlsanity.com/paper/2112.09858/) (2021)
- [Exploring Complex Dynamical Systems via Nonconvex Optimization](https://mlsanity.com/paper/2301.00923/) (2023)

## Abstract (arXiv)

We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with an underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Short-Term Memory (LSTM) network. It is shown that, when the numerical discretization is used, the function estimation problem can be solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose a smoothness condition in the prediction of the probability distribution. We show that the time evolution of the probability distribution can be computed by a high-dimensional integration of the transition probability of the LSTM internal states. A Monte Carlo algorithm to approximate the high-dimensional integration is outlined. The behavior of DE-LSTM is thoroughly investigated by using the Ornstein-Uhlenbeck process and noisy observations of nonlinear dynamical systems; Mackey-Glass time series and forced Van der Pol oscillator. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the stochastic process. For a multiple-step forecast of the Mackey-Glass time series, the prediction uncertainty, denoted by the 95% confidence interval, first grows, then dynamically adjusts following the evolution of the system, while in the simulation of the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for a 3,000-step forecast.