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

TL;DR

This paper introduces a novel non-linear reduction technique for Hamiltonian systems using convolutional auto-encoders and Hamiltonian neural networks, improving over traditional linear methods in preserving stability and dynamics.

Contribution

It presents a coupled training approach for auto-encoders and Hamiltonian neural networks to achieve effective non-linear model reduction for PDE discretizations.

Findings

Outperforms standard linear Hamiltonian reduction methods

Preserves long-term stability properties of the original system

Effective on non-linear wave dynamics test cases

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.