Learning Generalized Hamiltonians using fully Symplectic Mappings

2024-09-17

arXiv.org

AI Summary

This work addresses modeling and long-term prediction of non-separable Hamiltonian systems. We propose a physics-informed learning framework that rigorously preserves both the symplectic geometric structure and energy conservation. Methodologically, we generalize fully symplectic mappings to general non-separable Hamiltonian systems for the first time and design self-adjoint symplectic integrators to bypass backpropagation through ODE solvers—thereby substantially improving training efficiency and noise robustness. Coupled with physics-constrained neural network parameterization, our approach directly learns the generalized Hamiltonian function from noisy state observations. Experiments demonstrate that our method significantly outperforms existing Hamiltonian Neural Networks (HNNs) in Hamiltonian reconstruction accuracy, long-term simulation stability, and energy conservation: notably, energy error is reduced by over an order of magnitude on non-separable systems. This establishes a new paradigm for identification and prediction of complex conservative dynamical systems.

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Abstract

Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and in particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model. By ensuring physical invariances are conserved, the models exhibit significantly better sample complexity and out-of-distribution accuracy than standard NNs. Learning the Hamiltonian as a function of its canonical variables, typically position and velocity, from sample observations of the system thus becomes a critical task in system identification and long-term prediction of system behavior. However, to truly preserve the long-run physical conservation properties of Hamiltonian systems, one must use symplectic integrators for a forward pass of the system's simulation. While symplectic schemes have been used in the literature, they are thus far limited to situations when they reduce to explicit algorithms, which include the case of separable Hamiltonians or augmented non-separable Hamiltonians. We extend it to generalized non-separable Hamiltonians, and noting the self-adjoint property of symplectic integrators, we bypass computationally intensive backpropagation through an ODE solver. We show that the method is robust to noise and provides a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation. In the numerical results, we show the performance of the method concerning Hamiltonian reconstruction and conservation, indicating its particular advantage for non-separable systems.

Problem

• Learning generalized non-separable Hamiltonians from noisy observations
• Preserving long-run physical conservation properties with symplectic integrators
• Improving Hamiltonian reconstruction and conservation for system identification

Innovation

• Symplectic mappings for generalized Hamiltonians
• Bypassing backpropagation via self-adjoint property
• Robust Hamiltonian learning from noisy observations