Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lagrangian Neural Networks for Reversible Dissipative Evolution

Veera Sundararaghavan, Megna N. Shah, Jeff P. Simmons

TL;DR

The paper addresses the challenge of reversing dissipative dynamics by extending Lagrangian mechanics with Morse-Feshbach coordinate doubling to create a conservative augmented system. It introduces a Dissipative Lagrangian $\,mathcal{D}$ that depends only on observables, and trains a neural network to learn the form of $\,mathcal{D}$, enabling forward and reverse evolution for both mechanical and diffusion processes. Key contributions include generalizing Morse-Feshbach to multidimensional settings, deriving observable-based learning rules, and demonstrating accurate, invertible dynamics on a four-DOF mechanical system and multi-pixel Fickian diffusion with extrapolatory and inverse-diffusion capabilities. This approach broadens physics-informed learning to dissipative phenomena, offering a versatile framework for real-world materials science problems where reversibility and fast, regularized time evolution are valuable.

Abstract

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.

Lagrangian Neural Networks for Reversible Dissipative Evolution

TL;DR

The paper addresses the challenge of reversing dissipative dynamics by extending Lagrangian mechanics with Morse-Feshbach coordinate doubling to create a conservative augmented system. It introduces a Dissipative Lagrangian that depends only on observables, and trains a neural network to learn the form of , enabling forward and reverse evolution for both mechanical and diffusion processes. Key contributions include generalizing Morse-Feshbach to multidimensional settings, deriving observable-based learning rules, and demonstrating accurate, invertible dynamics on a four-DOF mechanical system and multi-pixel Fickian diffusion with extrapolatory and inverse-diffusion capabilities. This approach broadens physics-informed learning to dissipative phenomena, offering a versatile framework for real-world materials science problems where reversibility and fast, regularized time evolution are valuable.

Abstract

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.
Paper Structure (16 sections, 20 equations, 7 figures, 1 table)

This paper contains 16 sections, 20 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Preliminary Concepts
  5. Learning Lagrangians for Dissipative Problems
  6. Lagrangian Formulation of Dissipative Mechanics
  7. Modeling Diffusive Systems
  8. Neural network architecture
  9. Experiments
  10. Mechanics
  11. Fickian Diffusion
  12. Two Pixel
  13. Comparison to baseline model
  14. Sequence of diffusion images
  15. Inverse diffusion following Lagrangian trajectory
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Neural network architectures for (a) diffusion problem and (b) the dissipative mechanics problems
  • Figure 2: 4-D phase space of the mass--spring--damper system, projected onto the the 3-D space of $(x_1,x_2,\dot{x}_1)$. (a) data used for training containing 16 trajectories (b) DLNN prediction for 625 new trajectories.
  • Figure 3: (a) Training data containing 16 trajectories in phase space (b) Predictions of DLNN for the test set in phase space and (c) concentration versus time upto t =6 s.
  • Figure 4: (a) Results from the baseline and DLNN model are compared for an extrapolatory case. (b) The differences between the DLNN and baseline model zoomed in beyond $t=4.5$ seconds
  • Figure 5: (a) Microstructure model with simulation parameters (b) (top) DLNN simulation at different time steps showing concentration profiles compared to FEM results (bottom)
  • ...and 2 more figures