The time evolution of physical systems is described by differential
equations, which depend on abstract quantities like energy and force.
Traditionally, these quantities are derived as functionals based on observables
such as positions and velocities. Discovering these governing symbolic laws is
the key to comprehending the interactions in nature. Here, we present a
Hamiltonian graph neural network (HGNN), a physics-enforced GNN that learns the
dynamics of systems directly from their trajectory. We demonstrate the
performance of HGNN on n-springs, n-pendulums, gravitational systems, and
binary Lennard Jones systems; HGNN learns the dynamics in excellent agreement
with the ground truth from small amounts of data. We also evaluate the ability
of HGNN to generalize to larger system sizes, and to hybrid spring-pendulum
system that is a combination of two original systems (spring and pendulum) on
which the models are trained independently. Finally, employing symbolic
regression on the learned HGNN, we infer the underlying equations relating the
energy functionals, even for complex systems such as the binary Lennard-Jones
liquid. Our framework facilitates the interpretable discovery of interaction
laws directly from physical system trajectories. Furthermore, this approach can
be extended to other systems with topology-dependent dynamics, such as cells,
polydisperse gels, or deformable bodies.

利用哈密尔顿图神经网络 (HGNN) 直接从物理系统轨迹学习系统动力学，推断能量泛函的根本方程，并从物理系统轨迹中透明地发现相互作用定律。

用 Hamiltonian 图神经网络直接从轨迹中发现符号规律

Discovering Symbolic Laws Directly from Trajectories with Hamiltonian  Graph Neural Networks

We identify a set of "energy" functionals on the space of metrics in a given
Kaehler class on a Calabi-Yau manifold, which are bounded below and minimized
uniquely on the Ricci-flat metric in that class. Using these functionals, we
recast the problem of numerically solving the Einstein equation as an
optimization problem. We apply this strategy, using the "algebraic" metrics
(metrics for which the Kaehler potential is given in terms of a polynomial in
the projective coordinates), to the Fermat quartic and to a one-parameter
family of quintics that includes the Fermat and conifold quintics. We show that
this method yields approximations to the Ricci-flat metric that are
exponentially accurate in the degree of the polynomial (except at the conifold
point, where the convergence is polynomial), and therefore orders of magnitude
more accurate than the balanced metrics, previously studied as approximations
to the Ricci-flat metric. The method is relatively fast and easy to implement.
On the theoretical side, we also show that the functionals can be used to give
a heuristic proof of Yau's theorem.

利用特定空间度量上的一个 “能量” 函数，将数值求解爱因斯坦方程转化为优化问题，并且证明了该函数可以用于 Yau 定理的启发性证明。