Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks

2025-08-26

AI Summary

Existing MLP-based Hamiltonian Neural Networks (HNNs) suffer from high hyperparameter sensitivity, significant energy drift, and poor long-term prediction stability under complex energy landscapes. To address these issues, we propose KAR-HNN—a structured, low-degree-of-freedom energy function approximator grounded in the Kolmogorov–Arnold representation theorem, which replaces conventional multilayer perceptrons with learnable univariate functions. KAR-HNN integrates symplectic integration for gradient-based optimization, preserving the underlying symplectic geometry and ensuring physical interpretability while substantially reducing hyperparameter dependence. Experiments on spring-mass, pendulum, and two- and three-body systems demonstrate that KAR-HNN markedly suppresses energy drift and achieves high-accuracy, robust long-term dynamical modeling—even in high-dimensional, parameter-scarce regimes. Moreover, it captures both high-frequency responses and multi-scale dynamics effectively.

Technology Category

• Machine Learning: Kernel Methods
• Humans and AI: Human-Aware Planning and Behavior Prediction
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Large language models for search
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

We propose a Kolmogorov-Arnold Representation-based Hamiltonian Neural Network (KAR-HNN) that replaces the Multilayer Perceptrons (MLPs) with univariate transformations. While Hamiltonian Neural Networks (HNNs) ensure energy conservation by learning Hamiltonian functions directly from data, existing implementations, often relying on MLPs, cause hypersensitivity to the hyperparameters while exploring complex energy landscapes. Our approach exploits the localized function approximations to better capture high-frequency and multi-scale dynamics, reducing energy drift and improving long-term predictive stability. The networks preserve the symplectic form of Hamiltonian systems, and thus maintain interpretability and physical consistency. After assessing KAR-HNN on four benchmark problems including spring-mass, simple pendulum, two- and three-body problem, we foresee its effectiveness for accurate and stable modeling of realistic physical processes often at high dimensions and with few known parameters.

Problem

• Improving Hamiltonian Neural Networks' stability and accuracy
• Reducing energy drift in complex physical system modeling
• Enhancing predictive performance for high-dimensional dynamics

Innovation

• Replaces MLPs with univariate transformations
• Uses localized function approximations for dynamics
• Preserves symplectic form for physical consistency