Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation laws

Accurately finding and predicting dynamics based on the observational data with noise perturbations is of paramount significance but still a major challenge presently. Here, for the Hamiltonian mechanics, we propose the Hamiltonian neural Koopman operator (HNKO), integrating the knowledge of mathematical physics in learning the Koopman operator, and making it automatically sustain and even discover the conservation laws. We demonstrate the outperformance of the HNKO and its extension using a number of representative physical systems even with hundreds or thousands of freedoms. Our results suggest that feeding the prior knowledge of the underlying system and the mathematical theory appropriately to the learning framework can reinforce the capability of machine learning in solving physical problems.

Figure 1

A sketch for the HNKO framework. (a) A combination of the autoencoder with the orthogonal Koopman matrix $\mathit{K}$ using NNs. (b) Geometrically, the encoder embeds the original data in ${\mathbb{R}}^n$ to some low-dimensional manifold on $p$-dimensional sphere, and the decoder reverses this process, where $\mathit{K}$ maps the trajectory on the embedded manifold.

Figure 2

Comparison studies on the three-body problem. (a) The original and noise-free dynamics for the interacted bodies. Here, the motion ${\mathit{q}}_i=(q_i^1,q_i^2)$, and the trajectories are the projections from the original spatiotemporal space $q_i^1-q_i^2-t$ to the phase plane with a normal vector as $[sin(-\frac{\pi }{50})cos\frac{\pi }{4},sin(-\frac{\pi }{50})sin\frac{\pi }{4},cos(-\frac{\pi }{50})]$. The gray direct line indicates the time direction and the terminal positions of the three bodies are highlighted by blue, purple, and orange colors, respectively. The reconstructed and the predicted dynamics using the HNKO (b) and the other the most advanced machine learning techniques (c)–(i) are shown, respectively. (j) The temporal variance in the logarithm scale $[log(var\left)\right]$ of the features and the discovered Hamiltonian. (k) The system's energies using different methods change over the time. Here, we set $m_{1,2,3}=g=1$.

Figure 3

Comparison studies on the Kepler problem. (a) The original dynamics and the predicted phase orbits using different methods and the coordinate $\mathit{q}=(q^1,q^2)$. The changes of the kinetic energy $E_k$ (b), the potential energy $E_p$ (c), and the total energy $E$ (d) over the time for different methods. (e) Prediction errors of the state and the energy change with the noise variance ${\sigma }^2$, using HNKO. Here, $m=g=1$. (f) $log\left(var\right)$ of the features and the discovered Hamiltonian.

Figure 4

Comparison studies on the stiff mass-spring system. (a) The mean-square error (${\mathrm{MSE}}_{\text{data}}$) between the normalized trajectories and the vertical line segment under different SCs. The subfigures show the trajectories in the least and the stiffest cases. (b) The mean prediction MSE on time interval [0,9] of different methods over SCs. The prediction error (c) and the energy (d) over the time for different methods.

Figure 5

Comparison studies on the KdV equation. (a) The dynamics produced by the KdV equation and perturbed with the Gaussian noise $\mathcal{N}(0,0.03\mathit{I})$. The predicted trajectories using the DMD (b), the NODE (c), and the HNKO (d). Different solitary waves produced by using different methods at $t=400$ (e), the prediction error for the state (f), the total mass (g), and the total energy (h) over the time using different methods. The inset in (g) zoom in the changes of mass using the NODE and the HNKO. Here, we introduce the discretization to $\mathit{u}(t,·)$ on [0,50] by 64 predefined grid points.