Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems

Abstract

We propose an effective and lightweight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At the heart of our algorithm is a novel neural network architecture consisting of two sub-networks. Both are embedded with terms in the form of Taylor series expansion designed with symmetric structure. The key mechanism underpinning our infrastructure is the strong expressiveness and special symmetric property of the Taylor series expansion, which naturally accommodate the numerical fitting process of the gradients of the Hamiltonian with respect to the generalized coordinates as well as preserve its symplectic structure. We further incorporate a fourth-order symplectic integrator in conjunction with neural ODEs' framework into our Taylor-net architecture to learn the continuous-time evolution of the target systems while simultaneously preserving their symplectic structures. We demonstrated the efficacy of our Taylor-net in predicting a broad spectrum of Hamiltonian dynamic systems, including the pendulum, the Lotka--Volterra, the Kepler, and the Hénon--Heiles systems. Our model exhibits unique computational merits by outperforming previous methods to a great extent regarding the prediction accuracy, the convergence rate, and the robustness despite using extremely small training data with a short training period (6000 times shorter than the predicting period), small sample sizes, and no intermediate data to train the networks.

References in corpus (12)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Interaction Networks for Learning about Objects, Relations and Physics
• Inferring solutions of differential equations using noisy multi-fidelity data
• Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions
• Structure-preserving neural networks
• Hamiltonian Graph Networks with ODE Integrators
• Lagrangian Neural Networks
• Deep Hamiltonian networks based on symplectic integrators
• Sparse Symplectically Integrated Neural Networks
• Inverse modified differential equations for discovery of dynamics
• RoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data
• Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics

Cited by in corpus (16)

• Deep learning of thermodynamics-aware reduced-order models from data
• Symplectic Learning for Hamiltonian Neural Networks
• Physics-consistent machine learning: output projection onto physical manifolds
• Locally-symplectic neural networks for learning volume-preserving dynamics
• Machine learning structure preserving brackets for forecasting irreversible processes
• Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling
• Machine-learning Kohn-Sham potential from dynamics in time-dependent Kohn-Sham systems
• Inverse modified differential equations for discovery of dynamics
• Learning Potentials of Quantum Systems using Deep Neural Networks
• Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics
• Exploring explicit coarse-grained structure in artificial neural networks
• SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
• Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data
• Learning Trajectories of Hamiltonian Systems with Neural Networks
• SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision
• Nonseparable Symplectic Neural Networks