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Hamiltonian reduction using a convolutional auto-encoder coupled to an Hamiltonian neural network

Raphaël Côte, Emmanuel Franck, Laurent Navoret, Guillaume Steimer, Vincent Vigon

TL;DR

This paper proposes a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks, and has better reduction properties than standard linear Hamiltonian reduction methods.

Abstract

The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.

Hamiltonian reduction using a convolutional auto-encoder coupled to an Hamiltonian neural network

TL;DR

This paper proposes a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks, and has better reduction properties than standard linear Hamiltonian reduction methods.

Abstract

The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.
Paper Structure (32 sections, 69 equations, 19 figures, 8 tables)

This paper contains 32 sections, 69 equations, 19 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Parameterized Hamiltonian systems and reduction
  3. Parameterized Hamiltonian dynamics
  4. Hamiltonian reduced order modeling
  5. A non-linear Hamiltonian reduction method
  6. Reduction with an Auto Encoder (AE)
  7. Reduced model with a Hamiltonian Neural Network (HNN)
  8. Strong coupling of the neural networks
  9. Training hyper-parameters
  10. Reduced dimension $K$.
  11. Watch duration in predictions.
  12. Loss functions weights.
  13. Gradient descent.
  14. Pre-processing
  15. Numerical complexity
  16. ...and 17 more sections

Figures (19)

  • Figure 1: Prediction using the reduced model. The closed loop in the middle refers to the application of several iterations of the Störmer-Verlet scheme.
  • Figure 2: Auto-encoder architecture: encoder in blue, decoder in green. Symbols. $\ast$: stride 1 convolution with periodic padding, $\Downarrow$ down-sampling (stride 2 convolution), $\Uparrow$: up-sampling (repeat once each value along the last axis then smooth it with a kernel size $2$ convolution).
  • Figure 3: Example of loss functions history during a training, overlaid with the evolution of the learning rate (green).
  • Figure 4: (Linear wave) Solution $(\mathbf{u},\mathbf{v})$ at final time $T=0.4$ for various parameters $\mu=\mu_a$
  • Figure 5: (Linear wave) Solution $(\mathbf{u},\mathbf{v})$ at different times on test 3, reference solution (full lines) and prediction (dashed lines).
  • ...and 14 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2