CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems

2026-07-03

AI Summary

This work addresses the challenge of accurately learning the dissipative dynamics of linearly damped Hamiltonian systems from discrete observational data by proposing the CSympNet-ID framework. The method constructs neural networks that strictly preserve the discrete conformal symplectic structure, enabling direct learning of the one-step flow map from snapshot pairs without requiring penalty terms or projections. Its core innovation lies in integrating symplectic neural kernels with an exponentially parameterized diagonal scaling layer, which guarantees a positive-definite and physically interpretable dissipation factor. The authors further establish a scaling-conjugate decomposition theory for conformal symplectic maps. Experiments demonstrate that the proposed approach significantly outperforms existing baselines under data scarcity, accurately recovers contraction rates, and achieves state-of-the-art prediction accuracy and parameter identification performance across multiple damped oscillator systems, even in high-dimensional settings.

Abstract

Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID), a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregular-step damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.

Problem

• dissipative dynamics
• linearly damped Hamiltonian systems
• conformal symplecticity
• discrete observations
• long-horizon prediction

Innovation

• conformal symplectic
• dissipative dynamics
• symplectic neural networks
• damping identification
• structure-preserving learning