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

2023/11/10 by Raphaël Côte, Emmanuel Franck, Côte, Raphaël +7 · 3 citations
Engineering · Mathematics · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Power System Optimization and Stability

paper · pdf · doi:10.48550/arxiv.2311.06104

openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Citations

• Symplectic Model Reduction of Hamiltonian Systems
• Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
• Hamiltonian Neural Networks
• Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism
• Structure-Preserving Galerkin POD-DEIM Reduced-Order Modeling of Hamiltonian Systems

Cited by

• Neural non-canonical Hamiltonian dynamics for long-time simulations
• GeoHNNs: Geometric Hamiltonian Neural Networks
• Reduced Particle in Cell method for the Vlasov-Poisson system using auto-encoder and Hamiltonian neural

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