[Paper Review] Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings

Süleyman Yıldız, Pawan Goyal+2 more|arXiv (Cornell University)|Aug 2, 2023

Model Reduction and Neural NetworksPhysics and Astronomy3 citations

TL;DR

This paper proposes a data-driven framework to identify quadratic Hamiltonian systems for nonlinear dynamical systems by lifting the state space using weakly symplectic autoencoders. It enables structure-preserving, low-complexity modeling with long-term stability, achieving accurate dynamics in both low- and high-dimensional systems through learned cubic Hamiltonians in transformed coordinates.

ABSTRACT

This repository contains the Python implementation using the PyTorch framework to learn Quadratic Hamiltonian systems.

Motivation & Objective

• To develop a data-driven method for identifying quadratic Hamiltonian systems from trajectory data without requiring knowledge of the original equations.
• To enforce Hamiltonian structure in a lifted coordinate system while minimizing model complexity through a cubic Hamiltonian formulation.
• To enable dimensionality reduction for high-dimensional Hamiltonian systems by learning a low-dimensional latent dynamics with preserved symplectic structure.
• To ensure long-term stability and accurate extrapolation in unseen initial conditions through structure-preserving learning.
• To combine weakly enforced symplectic transformations with strongly enforced Hamiltonian dynamics for non-intrusive, nonlinear model order reduction.

Proposed method

• Lift the original state space into a higher-dimensional coordinate system using a neural network-based autoencoder, where the dynamics become quadratic.
• Enforce the symplectic structure of the transformation weakly via a regularization term in the loss function, ensuring the mapping preserves the symplectic form approximately.
• Learn a cubic Hamiltonian function in the lifted space, which gives rise to quadratic dynamics, enabling low-complexity modeling.
• Use a deep neural network architecture with skip connections and SELU activations to improve training stability and reconstruction accuracy.
• Train the autoencoder end-to-end with a composite loss combining reconstruction error, symplecticity regularization, and dynamics consistency via Hamiltonian structure enforcement.
• For high-dimensional data, apply the same framework to learn a reduced latent space, achieving efficient, structure-preserving model order reduction.

Figure 3.1 : The auto-encoder structure of the symplectic lifting method. Here, the encoder $\psi:\mathbb{R}^{2n}\to\mathbb{R}^{2N}$ is weakly enforced to be a symplectic mapping and the quadratic system is enforced to be Hamiltonian.

Experimental results

Research questions

• RQ1 Can nonlinear Hamiltonian systems be effectively represented as quadratic systems in a lifted coordinate space?
• RQ2 How can symplectic structure be weakly enforced in a data-driven autoencoder to preserve geometric properties of the dynamics?
• RQ3 Can a cubic Hamiltonian in the lifted space yield a low-complexity, stable quadratic dynamical system?
• RQ4 To what extent can this framework enable accurate long-term extrapolation of trajectories from unseen initial conditions?
• RQ5 How effective is the method for high-dimensional Hamiltonian systems, particularly in reducing dimensionality while preserving structure?

Key findings

• The method successfully learns quadratic Hamiltonian dynamics from data, achieving long-term stability in extrapolation beyond the training horizon.
• For low-dimensional systems, the framework identifies a 4-dimensional lifted space where the dynamics are quadratic and the Hamiltonian is cubic, ensuring low model complexity.
• In high-dimensional systems such as the nonlinear Schrödinger equation and wave equation, the method learns a 2- or 4-dimensional latent space that preserves the Hamiltonian structure and enables accurate dynamics prediction.
• The weakly symplectic autoencoder achieves balanced loss reduction across reconstruction, symplecticity, and dynamics consistency, with hyperparameters tuned experimentally.
• The model generalizes well to unseen initial conditions, demonstrating robustness in trajectory prediction under similar data distributions.
• The approach enables non-intrusive, structure-preserving model order reduction by learning latent dynamics directly from data without access to the original system equations.

Figure 4.1 : The auto-encoder structure of the symplectic reduction method. Here, the decoder $\phi:\mathbb{R}^{2n}\to\mathbb{R}^{2N}$ is weakly enforced to be a symplectic mapping and the quadratic system is enforced to be Hamiltonian.