# Computing non-equilibrium trajectories by a deep learning approach

arXiv: https://arxiv.org/abs/2210.04042 · HTML: https://mlsanity.com/paper/2210.04042/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 77 references that are in the map.
- First posted on arXiv 8 Oct 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Multi-scale Deep Neural Networks for Solving High Dimensional PDEs](https://mlsanity.com/paper/1910.11710/) (2019)
- [Chasing Collective Variables using Autoencoders and biased trajectories](https://mlsanity.com/paper/2104.11061/) (2021)
- [Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks](https://mlsanity.com/paper/2110.14523/) (2021)
- [An augmented Lagrangian deep learning method for variational problems with essential boundary conditions](https://mlsanity.com/paper/2106.14348/) (2021)

## Built on by (most cited first, 3 of 3)

- [Generative Path-Finding Method for Wasserstein Gradient Flow](https://mlsanity.com/paper/2604.11519/) (2026)
- [StringNET: Neural Network based Variational Method for Transition Pathways](https://mlsanity.com/paper/2408.12621/) (2024)
- [ASPEN: An Adaptive Spectral Physics-Enabled Network for Ginzburg-Landau Dynamics](https://mlsanity.com/paper/2512.03290/) (2025)

## Abstract (arXiv)

Predicting the occurence of rare and extreme events in complex systems is a well-known problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in the last ten years, which better estimate tail distributions. They often use large deviation concepts without the need to perform heavy direct ensemble simulations. In particular, a well-known approach is to derive a minimum action principle and to find its minimizers. The analysis of rare reactive events in non-equilibrium systems without detailed balance is notoriously difficult either theoretically and computationally. They are described in the limit of small noise by the Freidlin-Wentzell action. We propose here a new method which minimizes the geometrical action instead using neural networks: it is called deep gMAM. It relies on a natural and simple machine-learning formulation of the classical gMAM approach. We give a detailed description of the method as well as many examples. These include bimodal switches in complex stochastic (partial) differential equations, quasi-potential estimates, and extreme events in Burgers turbulence.