# An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps

arXiv: https://arxiv.org/abs/2203.16874 · HTML: https://mlsanity.com/paper/2203.16874/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 59 references that are in the map.
- First posted on arXiv 31 Mar 2022.

## Builds on (most cited first, 8 of 8)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks](https://mlsanity.com/paper/1803.01299/) (2018)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [Transition path theory for Langevin dynamics on manifold: optimal control and data-driven solver](https://mlsanity.com/paper/2010.09988/) (2020)

## Built on by (most cited first, 2 of 2)

- [Early Warning Prediction with Automatic Labeling in Epilepsy Patients](https://mlsanity.com/paper/2310.06059/) (2023)
- [StringNET: Neural Network based Variational Method for Transition Pathways](https://mlsanity.com/paper/2408.12621/) (2024)

## Abstract (arXiv)

Many complex real world phenomena exhibit abrupt, intermittent or jumping behaviors, which are more suitable to be described by stochastic differential equations under non-Gaussian L'evy noise. Among these complex phenomena, the most likely transition paths between metastable states are important since these rare events may have a high impact in certain scenarios. Based on the large deviation principle, the most likely transition path could be treated as the minimizer of the rate function upon paths that connect two points. One of the challenges to calculate the most likely transition path for stochastic dynamical systems under non-Gaussian L'evy noise is that the associated rate function can not be explicitly expressed by paths. For this reason, we formulate an optimal control problem to obtain the optimal state as the most likely transition path. We then develop a neural network method to solve this issue. Several experiments are investigated for both Gaussian and non-Gaussian cases.