# Lagrangian Neural Networks for Reversible Dissipative Evolution

arXiv: https://arxiv.org/abs/2405.14645 · HTML: https://mlsanity.com/paper/2405.14645/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 22 references that are in the map.
- First posted on arXiv 23 May 2024.

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

- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)

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

- [Floating-Body Hydrodynamic Neural Networks](https://mlsanity.com/paper/2509.13783/) (2025)
- [Lagrangian and Hamiltonian Neural Networks With a Dissipative System](https://mlsanity.com/paper/2609.31988/) (2026)

## Abstract (arXiv)

There is a growing attention given to utilizing Lagrangian and Hamiltonian mechanics with network training in order to incorporate physics into the network. Most commonly, conservative systems are modeled, in which there are no frictional losses, so the system may be run forward and backward in time without requiring regularization. This work addresses systems in which the reverse direction is ill-posed because of the dissipation that occurs in forward evolution. The novelty is the use of Morse-Feshbach Lagrangian, which models dissipative dynamics by doubling the number of dimensions of the system in order to create a mirror latent representation that would counterbalance the dissipation of the observable system, making it a conservative system, albeit embedded in a larger space. We start with their formal approach by redefining a new Dissipative Lagrangian, such that the unknown matrices in the Euler-Lagrange's equations arise as partial derivatives of the Lagrangian with respect to only the observables. We then train a network from simulated training data for dissipative systems such as Fickian diffusion that arise in materials sciences. It is shown by experiments that the systems can be evolved in both forward and reverse directions without regularization beyond that provided by the Morse-Feshbach Lagrangian. Experiments of dissipative systems, such as Fickian diffusion, demonstrate the degree to which dynamics can be reversed.