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Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings

Süleyman Yildiz, Pawan Goyal, Thomas Bendokat, Peter Benner

TL;DR

This work tackles learning Hamiltonian dynamics from data while preserving the intrinsic energy structure. It introduces a lifting principle that maps nonlinear Hamiltonian systems to quadratic dynamics in a latent space, enforcing a cubic latent Hamiltonian via a weakly symplectic auto-encoder and learning the latent, structure-preserving system through a quadratic form $\dot z = \mathcal{A} + \mathcal{B}z + \mathcal{C}(z \otimes z)$. The approach covers both lifting to high dimensions and reducing high-dimensional data to a low-dimensional quadratic representation, validated on classic low-dimensional systems and PDEs (including the linear wave and nonlinear Schrödinger equations), with long-time stability and energy conservation demonstrated. This framework yields non-intrusive, structure-preserving reduced-order models that bypass explicit gradient computations of the Hamiltonian and avoid hyper-reduction, while remaining applicable even when the full-order model is unavailable. The work also highlights practical considerations such as computational cost and suggests directions for handling noise, experimenting with different auto-encoder architectures, and extending to discrete or controlled Hamiltonian systems.

Abstract

We present a framework for learning Hamiltonian systems using data. This work is based on a lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this, we obtain quadratic dynamics that are Hamiltonian in a transformed coordinate system. To that end, for given generalized position and momentum data, we propose a methodology to learn quadratic dynamical systems, enforcing the Hamiltonian structure in combination with a weakly-enforced symplectic auto-encoder. The obtained Hamiltonian structure exhibits long-term stability of the system, while the cubic Hamiltonian function provides relatively low model complexity. For low-dimensional data, we determine a higher-dimensional transformed coordinate system, whereas for high-dimensional data, we find a lower-dimensional coordinate system with the desired properties. We demonstrate the proposed methodology by means of both low-dimensional and high-dimensional nonlinear Hamiltonian systems.

Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings

TL;DR

This work tackles learning Hamiltonian dynamics from data while preserving the intrinsic energy structure. It introduces a lifting principle that maps nonlinear Hamiltonian systems to quadratic dynamics in a latent space, enforcing a cubic latent Hamiltonian via a weakly symplectic auto-encoder and learning the latent, structure-preserving system through a quadratic form . The approach covers both lifting to high dimensions and reducing high-dimensional data to a low-dimensional quadratic representation, validated on classic low-dimensional systems and PDEs (including the linear wave and nonlinear Schrödinger equations), with long-time stability and energy conservation demonstrated. This framework yields non-intrusive, structure-preserving reduced-order models that bypass explicit gradient computations of the Hamiltonian and avoid hyper-reduction, while remaining applicable even when the full-order model is unavailable. The work also highlights practical considerations such as computational cost and suggests directions for handling noise, experimenting with different auto-encoder architectures, and extending to discrete or controlled Hamiltonian systems.

Abstract

We present a framework for learning Hamiltonian systems using data. This work is based on a lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this, we obtain quadratic dynamics that are Hamiltonian in a transformed coordinate system. To that end, for given generalized position and momentum data, we propose a methodology to learn quadratic dynamical systems, enforcing the Hamiltonian structure in combination with a weakly-enforced symplectic auto-encoder. The obtained Hamiltonian structure exhibits long-term stability of the system, while the cubic Hamiltonian function provides relatively low model complexity. For low-dimensional data, we determine a higher-dimensional transformed coordinate system, whereas for high-dimensional data, we find a lower-dimensional coordinate system with the desired properties. We demonstrate the proposed methodology by means of both low-dimensional and high-dimensional nonlinear Hamiltonian systems.
Paper Structure (17 sections, 2 theorems, 34 equations, 19 figures, 5 tables)

This paper contains 17 sections, 2 theorems, 34 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Hamiltonian Systems and Symplectic Embedding
  4. Quadratic Hamiltonian Representations
  5. Learning the Lifted Quadratic Symplectic Representation
  6. Low-dimensional Quadratic Symplectic Representation of High-dimensional Data
  7. Numerical Experiments
  8. Low-dimensional Systems
  9. Nonlinear Pendulum
  10. Lotka–Volterra Equations
  11. Nonlinear Oscillator
  12. Wooden Pendulum Clock
  13. High-dimensional Systems
  14. Linear Wave Equation
  15. Nonlinear Schrödinger Equation
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\psi \colon \mathbb{R}^{2n} \to \mathbb{R}^{2N}$ be a symplectic embedding and define $z_0 := \psi(x_0) \in \mathbb{R}^{2N}$. Then, the system eq:HamiltonianEquations is equivalent to the embedded system in the sense that $z(t) := \psi(x(t))$ solves eq:EmbeddedSymplecticSystem for all $t \in [0,\infty)$.

Figures (19)

  • 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.
  • 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.
  • Figure 5.1: Nonlinear pendulum: Plot (a) shows the training data with the blue dots in phase space together with the ground truth vector field, and Plot (b) shows a comparison of the learned model \ref{['quadraticODEHamiltonian']} with the ground truth in phase space with three random initial conditions. Plot (c) shows a comparison of the learned deep linear embedding with the ground truth in phase space with three random initial test conditions.
  • Figure 5.2: Nonlinear pendulum: A comparison of the learned model with the ground truth model for a random test condition.
  • Figure 5.3: Nonlinear pendulum: A comparison of the Hamiltonian in canonical coordinates for the ground truth model $\mathcal{H}(q,p)$, the learned Hamiltonian $\hat{\mathcal{H}}(\hat{q}, \hat{p})$ in the latent space, and the difference between the ground truth model and the learned model in the original space $\mathcal{H}(q(\hat{q}, \hat{p}), p(\hat{q}, \hat{p}))$ along time using a random test initial condition.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Definition 1: Symplectic Embedding for Vector Spaces
  • Proposition 1: Equivalent Embedded System
  • proof
  • Example 1: Nonlinear Oscillator
  • Definition 2: Quadratic Hamiltonian System
  • Proposition 2
  • proof