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tLaSDI: Thermodynamics-informed latent space dynamics identification

Jun Sur Richard Park, Siu Wun Cheung, Youngsoo Choi, Yeonjong Shin

TL;DR

A latent space dynamics identification method, namely tLaSDI, that embeds the first and second principles of thermodynamics in the latent space dynamics identification method, which exhibits robust generalization ability, even in extrapolation.

Abstract

We propose a latent space dynamics identification method, namely tLaSDI, that embeds the first and second principles of thermodynamics. The latent variables are learned through an autoencoder as a nonlinear dimension reduction model. The latent dynamics are constructed by a neural network-based model that precisely preserves certain structures for the thermodynamic laws through the GENERIC formalism. An abstract error estimate is established, which provides a new loss formulation involving the Jacobian computation of autoencoder. The autoencoder and the latent dynamics are simultaneously trained to minimize the new loss. Computational examples demonstrate the effectiveness of tLaSDI, which exhibits robust generalization ability, even in extrapolation. In addition, an intriguing correlation is empirically observed between a quantity from tLaSDI in the latent space and the behaviors of the full-state solution.

tLaSDI: Thermodynamics-informed latent space dynamics identification

TL;DR

A latent space dynamics identification method, namely tLaSDI, that embeds the first and second principles of thermodynamics in the latent space dynamics identification method, which exhibits robust generalization ability, even in extrapolation.

Abstract

We propose a latent space dynamics identification method, namely tLaSDI, that embeds the first and second principles of thermodynamics. The latent variables are learned through an autoencoder as a nonlinear dimension reduction model. The latent dynamics are constructed by a neural network-based model that precisely preserves certain structures for the thermodynamic laws through the GENERIC formalism. An abstract error estimate is established, which provides a new loss formulation involving the Jacobian computation of autoencoder. The autoencoder and the latent dynamics are simultaneously trained to minimize the new loss. Computational examples demonstrate the effectiveness of tLaSDI, which exhibits robust generalization ability, even in extrapolation. In addition, an intriguing correlation is empirically observed between a quantity from tLaSDI in the latent space and the behaviors of the full-state solution.
Paper Structure (20 sections, 2 theorems, 37 equations, 8 figures, 4 tables)

This paper contains 20 sections, 2 theorems, 37 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Neural networks
  4. GENERIC formalism
  5. Thermodynamics-informed learning for latent space dynamics
  6. tLaSDI: Loss formulation
  7. Abstract error estimate
  8. Numerical examples
  9. Data-driven discovery of dynamical systems: Extrapolation
  10. Couette flow of an Oldroyd-B fluid
  11. Two gas containers exchanging heat and volume
  12. Parametric PDEs
  13. 1D Burgers' equation
  14. 1D heat equation
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 1

Let $A_j$ be a skew-symmetric matrix of size $n \times n$, $j = 1,\dots, m$ and $g(\cdot): \mathbb{R}^{n} \to \mathbb{R}$ a differentiable scalar function. For the matrix valued function $Q_g(\cdot): \mathbb{R}^{n} \to \mathbb{R}^{m \times n}$ whose $j$-th row is defined as $(A_j \nabla g(\cdot))^\t

Figures (8)

  • Figure 1: The schematic NN model of tLaSDI. The hyper-autoencoder is used for the parametric dynamical systems. The latent dynamics are modelled by GFINNs.
  • Figure 2: Example \ref{['sec:oldroyd']}. Left: The training loss trajectories for 10 simulations versus the wall time by the three methods. Right: The mean and one standard deviation away from the mean of the extrapolation errors \ref{['eq:rel_l2_nonpara']} versus the wall time. The vertical line indicates the time for TA-ROM trains SAE.
  • Figure 3: Example \ref{['sec:oldroyd']}. Four different GT trajectories and the corresponding predictions by tLaSDI (top row), TA-ROM (middle row), and Vanilla-FNN (bottom row). Each method uses the model with the smallest loss from $10$ independent simulations.
  • Figure 4: Example \ref{['sec:gas_containers']}. Left: The training loss trajectories for 10 simulations versus the wall time by the three methods. Right: The mean and one standard deviation away from the mean of the extrapolation errors \ref{['eq:rel_l2_nonpara']} versus the wall time. The vertical line indicates the time for TA-ROM trains SAE.
  • Figure 5: Example \ref{['sec:gas_containers']}: Four different GT trajectories and the corresponding predictions by tLaSDI (top row), TA-ROM (middle row), and Vanilla-FNN (bottom row). Each method uses the model with the smallest loss from $10$ independent simulations.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1: Lemma 3.6 of zhang2022gfinns
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof