Symplectic Recurrent Neural Networks

TL;DR

The paper tackles learning physical laws for Hamiltonian systems from observed trajectories, addressing numerical stiffness and noise. It introduces Symplectic Recurrent Neural Networks (SRNNs) that combine neural-network Hamiltonians with symplectic leapfrog integration, multi-step (recurrent) training, and initial-state optimization to improve robustness and accuracy. Empirical results on a 20-mass spring-chain and a chaotic three-body system show that SRNNs outperform non-symplectic baselines and can even surpass simulations using the true Hamiltonian by compensating for discretization errors. An augmented SRNN capable of handling perfect rebound demonstrates applicability to stiff dynamics, highlighting the framework’s potential for data-driven, structure-preserving numerical solvers in physics-informed machine learning.

Abstract

We propose Symplectic Recurrent Neural Networks (SRNNs) as learning algorithms that capture the dynamics of physical systems from observed trajectories. An SRNN models the Hamiltonian function of the system by a neural network and furthermore leverages symplectic integration, multiple-step training and initial state optimization to address the challenging numerical issues associated with Hamiltonian systems. We show that SRNNs succeed reliably on complex and noisy Hamiltonian systems. We also show how to augment the SRNN integration scheme in order to handle stiff dynamical systems such as bouncing billiards.

Figures (24)

• Figure 1: Testing results in the noiseless case by single-step methods. Left: Prediction error of each method over time, measured by the L2 distance between the true and predicted positions of the 20 masses. Right: Each curve represents the position of one of the masses (number 5) as a function of time predicted by the three single-step-trained H-NET models. Plots of the other masses' positions are provided in Appendix \ref{['additional_n0']}.
• Figure 2: Prediction error of all methods in the noisy case measured by L2 distance, presented in two plots due to the large number of methods. Included in the left plot are the single-step-trained methods, recurrently trained methods, vanilla RNN and LSTM. Included in the right plot are the (same) recurrently trained methods, the recurrently trained methods with initial state optimization (ISO), as well as vanilla RNN and LSTM with ISO.
• Figure 3: Predictions made by three methods in the noisy case. The Y-axis corresponds to the position of one of the masses (number 5) on the chain.
• Figure 4: Actual versus predicted trajectories of the heavy billiard with perfect rebound. The predictions are obtained by an SRNN plus the rebound module described in section \ref{['rebound']}.
• Figure 5: Extension of Figure \ref{['Fig:n0_1step']} to 10 masses on the chain (1st being the closest to one end, and 10th being in the center).