Constants of motion network revisited

TL;DR

The paper addresses the challenge of discovering constants of motion from dynamical data, especially in non-Hamiltonian systems and under noise. It introduces Meta-COMET, a singular-value-decomposition-inspired neural architecture with two parallel pathways for learning the initial state rate of change and constants of motion, combined with a two-phase training procedure that enforces semi-orthogonal constraints and optimizes dynamics fitting. Meta-COMET achieves substantial parameter reduction (e.g., up to ~14x fewer parameters in some cases) and improved noise robustness while retaining COMET's ability to identify constants of motion and handle external forces; it is validated across toy systems, forced dynamics, KdV, and pixel-based latent dynamics. The work has practical implications for efficient, coordinate-free discovery of invariants in diverse dynamical systems, enabling more robust scientific machine learning and physics-informed modeling.

Abstract

Discovering constants of motion is meaningful in helping understand the dynamical systems, but inevitably needs proficient mathematical skills and keen analytical capabilities. With the prevalence of deep learning, methods employing neural networks, such as Constant Of Motion nETwork (COMET), are promising in handling this scientific problem. Although the COMET method can produce better predictions on dynamics by exploiting the discovered constants of motion, there is still plenty of room to sharpen it. In this paper, we propose a novel neural network architecture, built using the singular-value-decomposition (SVD) technique, and a two-phase training algorithm to improve the performance of COMET. Extensive experiments show that our approach not only retains the advantages of COMET, such as applying to non-Hamiltonian systems and indicating the number of constants of motion, but also can be more lightweight and noise-robust than COMET.

Figures (10)

• Figure 1: The architecture of the two neural networks to calculate $\mathbf{\dot{s}_0}$ and $\mathbf{c}$. The inputs of each neural network are both the concatenation of the state $\mathbf{s}$ and the external force $\mathbf{F}$ (if present). The parameters of the four linear layers are all different. In each layer $l$, except for the additional parameters $V^l$, the $\textsc{svd-FC}^l$ layer and the $\textsc{sd-FC}^l$ layer have exactly the same $S ^l$ and $D^l$.
• Figure 2: With different $\sigma$ (equals to 0.05, 0.1, 0.2 from top to bottom), the contour plot of constant of motion discovered by Meta-COMET and COMET for the mass-spring case. The left column is for COMET. The middle column is for Meta-COMET. The last column is the ground truth.
• Figure 3: Same as Figure \ref{['fig:discovered-com1']} except for describing the Lotka-Volterra case.
• Figure 4: The evolution of constants of motion calculated by Meta-COMET and COMET for (a) mass-spring, (b) 2d pendulum, and (c) two body system. $\sigma$ is equal to 0.05, 0.1, and 0.2 from top to bottom in each case.
• Figure 5: From the left column to the right column, it represents the motion trajectory of the simulated two body system from $t=0$ to $t=20$ predicted by the COMET, Meta-COMET and its analytical equation when $\sigma = 0.05, 0.1, 0.2$ (from top to bottom).