Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach

2026-04-01

AI Summary

Existing neural network models struggle to accurately assess their fidelity in preserving the global geometric structures of Hamiltonian phase space, such as homoclinic orbits and separatrices. This work introduces Lagrangian descriptors (LDs) into this domain for the first time, embedding phase space geometry into an information-theoretic framework via a weighted probability density function to enable quantitative comparison of global dynamical structures across models. This approach overcomes the limitations of conventional trajectory-error-based evaluations. Experiments reveal that, in the Duffing oscillator, all tested models successfully reproduce homoclinic orbits; however, in a three-mode nonlinear Schrödinger system, unconstrained reservoir computing outperforms energy-conserving symplectic networks—including SympNet, HénonNet, and generalized Hamiltonian neural networks—in reconstructing homoclinic structures, highlighting a potential tension between topological fidelity and energy conservation.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

We propose Lagrangian Descriptors (LDs) as a diagnostic framework for evaluating neural network models of Hamiltonian systems beyond conventional trajectory-based metrics. Standard error measures quantify short-term predictive accuracy but provide little insight into global geometric structures such as orbits and separatrices. Existing evaluation tools in dissipative systems are inadequate for Hamiltonian dynamics due to fundamental differences in the systems. By constructing probability density functions weighted by LD values, we embed geometric information into a statistical framework suitable for information-theoretic comparison. We benchmark physically constrained architectures (SympNet, HénonNet, Generalized Hamiltonian Neural Networks) against data-driven Reservoir Computing across two canonical systems. For the Duffing oscillator, all models recover the homoclinic orbit geometry with modest data requirements, though their accuracy near critical structures varies. For the three-mode nonlinear Schrödinger equation, however, clear differences emerge: symplectic architectures preserve energy but distort phase-space topology, while Reservoir Computing, despite lacking explicit physical constraints, reproduces the homoclinic structure with high fidelity. These results demonstrate the value of LD-based diagnostics for assessing not only predictive performance but also the global dynamical integrity of learned Hamiltonian models.

Problem

• Hamiltonian dynamics
• phase space integrity
• Lagrangian descriptors
• neural network models
• geometric structure

Innovation

• Lagrangian Descriptors
• Hamiltonian dynamics
• neural network evaluation
• phase space integrity
• symplectic architectures