Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variational Neural Networks for Observable Thermodynamics (V-NOTS)

Christopher Eldred, François Gay-Balmaz, Vakhtang Putkaradze

Abstract

Much attention has recently been devoted to data-based computing of evolution of physical systems. In such approaches, information about data points from past trajectories in phase space is used to reconstruct the equations of motion and to predict future solutions that have not been observed before. However, in many cases, the available data does not correspond to the variables that define the system's phase space. We focus our attention on the important example of dissipative dynamical systems. In that case, the phase space consists of coordinates, momenta and entropies; however, the momenta and entropies cannot, in general, be observed directly. To address this difficulty, we develop an efficient data-based computing framework based exclusively on observable variables, by constructing a novel approach based on the thermodynamic Lagrangian, and constructing neural networks that respect the thermodynamics and guarantees the non-decreasing entropy evolution. We show that our network can provide an efficient description of phase space evolution based on a limited number of data points and a relatively small number of parameters in the system.

Variational Neural Networks for Observable Thermodynamics (V-NOTS)

Abstract

Much attention has recently been devoted to data-based computing of evolution of physical systems. In such approaches, information about data points from past trajectories in phase space is used to reconstruct the equations of motion and to predict future solutions that have not been observed before. However, in many cases, the available data does not correspond to the variables that define the system's phase space. We focus our attention on the important example of dissipative dynamical systems. In that case, the phase space consists of coordinates, momenta and entropies; however, the momenta and entropies cannot, in general, be observed directly. To address this difficulty, we develop an efficient data-based computing framework based exclusively on observable variables, by constructing a novel approach based on the thermodynamic Lagrangian, and constructing neural networks that respect the thermodynamics and guarantees the non-decreasing entropy evolution. We show that our network can provide an efficient description of phase space evolution based on a limited number of data points and a relatively small number of parameters in the system.

Paper Structure

This paper contains 37 sections, 3 theorems, 51 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background: general approaches to machine learning of systems with structure
  3. On equation discovery through data: a continuous approach.
  4. Metriplectic/GENERIC approach to describing dissipative systems.
  5. Equations discovery for metriplectic systems.
  6. Observable and non-observable data.
  7. Discovering equations vs mappings in phase space: continuous and discrete approaches.
  8. Differences from the metriplectic/GENERIC approach.
  9. Discussion on variational approaches for mechanical systems with friction.
  10. Difference from Onsager-based approaches.
  11. Problem statement: data-based computing of dissipative systems from observable data.
  12. Novelty of this paper.
  13. A short digression on systems with symmetry and observable variables.
  14. On the expression for the friction force.
  15. Background: Variational approach to dissipative systems and variational integrators
  16. ...and 22 more sections

Key Result

Lemma 3.2

Suppose a trajectory in the observable space $({\boldsymbol{q}}(t), {\boldsymbol{v}}(t), T(t))$ is a solution of EL_thermo with $G=G_*({\boldsymbol{q}}, {\boldsymbol{v}}, T)$, $\boldsymbol{F}^{\rm fr} =\boldsymbol{F}^{\rm fr}_*({\boldsymbol{q}}, {\boldsymbol{v}}, T)$ with some initial conditions. Th

Figures (6)

  • Figure 1: Illustration of the difference between evolution in observable and non-observable variables. The evolution equation \ref{['Dissipative_system_general_pq']} is written in terms of the phase-space variables ${\boldsymbol{u}} = ({\boldsymbol{q}},{\boldsymbol{p}},S)$, where the momentum ${\boldsymbol{p}}$ and entropy $S$ are not directly observable in experiments without additional information. In contrast, the observable data consist of the coordinates ${\boldsymbol{q}}$, velocities ${\boldsymbol{v}}$, and temperatures $T$. The friction force is also dependent on the observable variables, although it is acting as a part of equation \ref{['Brackets_metriplectic']}. The observable variables $({\boldsymbol{v}},T)$ and the non-observable variables $({\boldsymbol{p}},S)$ are related to each other by the derivatives of the Hamiltonian $H({\boldsymbol{q}},{\boldsymbol{p}},S)$, which contains both mechanical and thermal contributions. In Section \ref{['sec:observables']} we introduce a thermal Lagrangian $G({\boldsymbol{q}},{\boldsymbol{v}},T)$ which allows to connect the dynamics in observable variables $({\boldsymbol{q}},{\boldsymbol{v}},T)$ with the phase space description in terms of ${\boldsymbol{u}}$.
  • Figure 2: Setup of the problem of an adiabatic piston of mass $M$ separating two chambers. Each chamber $i=1,2$ has the area $A_{1,2}$ and the length $L \pm x$ where $x$ is the coordinate of the piston. Each of the chamber has gas with internal energy depending on the volume $V_i(x)$ and entropy of the gas $S_i$.
  • Figure 3: Trajectory reconstruction after learning when partial or minimal information about the system is available: either the information about $G$ or the force. Color notation for solid lines, consistent between both panel of this Figure and Figure \ref{['fig:error_energy_partially_known_GF']}. Black: ground truth. Purple: unknown $G$, known $F^{\rm fr}_{1,2}$. Green: known $G$, unknown $F^{\rm fr}_{1,2}$. Blue: both $G$ and $F^{\rm fr}_{1,2}$ are known, representing pure variational integrator. Red: neither $G$ nor $F^{\rm fr}_{1}$ are known. Orange: known $G$ and $F^{\rm fr}_{1}$, unknown $F^{\rm fr}_{2}$. Brown: known $G$ and $F^{\rm fr}_{2}$, unknown $F^{\rm fr}_{1}$. The red line represents the most challenging case, and yet neural network reconstructs the motion with roughly the same accuracy as the pure variational integrator with complete knowledge of the system.
  • Figure 4: Errors in energy (left panel) and MAE of individual components(right panel), with all the notations being exactly the same as Figure \ref{['fig:partially_known_GF']}. Energy is conserved with high precision and MAE of individual components(right panel) remains bounded. All color notations exactly as in Figure \ref{['fig:partially_known_GF']}.
  • Figure 5: Results of the learning scheme with various known and unknown quantities. Red line: unknown $G$, known $\boldsymbol{f}^{\rm fr}$. Green line: known $G$, unknown $\boldsymbol{f}^{\rm fr}$. Black line: both $\boldsymbol{f}^{\rm fr}$ and $G$ are known (variational integrator) which is the best outcome based on the discrete observable data. Orange line: the most challenging case when both $G$ and $\mathbf{f}^{\rm fr}$ are unknown. Blue line: ground truth. Top panel: non-observable variables: $(\mu_1, \mu_2, \mu_3, S)$. Bottom panel: observables $(\Omega_1, \Omega_2, \Omega_3, T)$. In this particular case, the temperature $T$ is proportional to the energy, so the conservation of temperature reflects the conservation of energy as illustrated on the left panel of Figure \ref{['fig:ThermoNets_Hamiltonian_learning_error']}.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 3.1
  • Lemma 3.2: On non-uniqueness of solutions
  • Remark 3.3: On non-uniqueness for the case of several temperatures
  • Theorem 4.1: Dissipative neural networks
  • Remark 4.2: On the Cholesky factorization of matrix S
  • Lemma 5.1: On the invariances of solutions
  • Remark 5.2
  • Remark 5.3: On the choice of linear term in ${\boldsymbol{v}}$