Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation

TL;DR

This work develops the method of the so-called neural deflation directly from system trajectories instead of using the explicit knowledge of the underlying equations of motion, which is crucial towards enhancing the practical implementation of the method in scenarios where solely data reflecting discrete snapshots of the system are available.

Abstract

In an earlier work by a subset of the present authors, the method of the so-called neural deflation was introduced towards identifying a complete set of functionally independent conservation laws of a nonlinear dynamical system. Here, we extend by a significant step this proposal. Instead of using the explicit knowledge of the underlying equations of motion, we develop the method directly from system trajectories. This is crucial towards enhancing the practical implementation of the method in scenarios where solely data reflecting discrete snapshots of the system are available. We showcase the results of the method and the number of associated conservation laws obtained in a diverse range of examples including 1D and 2D harmonic oscillators, the Toda lattice, the Fermi-Pasta-Ulam-Tsingou lattice and the Calogero-Moser system.

Figures (8)

• Figure 1: Schematic representation of our method. Trajectory data (with different colors representing distinct trajectories) are first input into the Hamiltonian Neural Network (HNN) to learn the Hamiltonian $H_{\bm{\xi}^*}(\mathbf{q}, \mathbf{p})$, which is then incorporated into the neural deflation method. A jump in the validation loss at $I_K$ signals the successful learning of $K-1$ Poisson-commuting, functionally independent conservation laws.
• Figure 2: Relative error of HNNs, as defined in Eq. \ref{['eq:relative_error']}, evaluated over the domain $[-10^4, 10^4]$ for the 1D harmonic oscillator.
• Figure 3: 2D harmonic oscillator. This figure shows the validation losses $\{\mathcal{L}_k(\bm{\theta}^{*}_k ; \mathcal{V})\}^{4}_{k=1}$ for the learned conserved quantities $\{I_k(\cdot; \bm{\theta}^{*}_k) \}^{4}_{k=1}$ under different deflation strengths ($\alpha=1.0$ and $\alpha=0.5$). Results are compared for: (a) our model, which learns solely from system trajectories, and (b) the original neural deflation method neural_deflation, which uses explicit knowledge of the ground truth differential equation. Both methods exhibit a significant increase in loss at $k=3$, indicating successful identification of the system's integrability, as per Algorithm \ref{['alg:deflation']}.
• Figure 4: Discrete sine-Gordon system with the number of lattice sites/degrees of freedom $N=d=6$. This figure shows the validation losses $\{\mathcal{L}_k(\bm{\theta}^{*}_k ; \mathcal{V})\}^{2d}_{k=1}$ for the learned conserved quantities $\{I_k(\cdot; \bm{\theta}^{*}_k) \}^{2d}_{k=1}$. Results are compared for: (a) our model, which learns solely from system trajectories, and (b) the original neural deflation method neural_deflation, which uses explicit knowledge of the ground truth differential equation. Both methods exhibit a significant increase in loss at $k=2$, consistent with the fact that the underlying system has only one independent conservation law; see Algorithm \ref{['alg:deflation']}.
• Figure 5: Calogero-Moser system with the number of lattice sites/degrees of freedom $N=d=6$. This figure shows the validation losses $\{\mathcal{L}_k(\bm{\theta}^{*}_k ; \mathcal{V})\}^{2d}_{k=1}$ for the learned conserved quantities $\{I_k(\cdot; \bm{\theta}^{*}_k) \}^{2d}_{k=1}$. Results are compared for: (a) our model, which learns solely from system trajectories, and (b) the original neural deflation method neural_deflation, which uses explicit knowledge of the ground truth differential equation. Both methods appear to exhibit a significant increase in loss at $k=7$, consistent with the fact that the underlying system is fully-integrable.