Discovering Symbolic Laws Directly from Trajectories with Hamiltonian
  Graph Neural Networks
Suresh Bishnoi Ravinder Bhattoo Jayadeva Sayan Ranu N M Anoop Krishnan
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.