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Recent years have witnessed an increased interest in recovering dynamicallaws of complex systems in a largely data-driven fashion under meaningfulhypotheses. In this work, we propose a method for scalably learning dynamicallaws of classical dynamical systems from data. As a novel ingredient, toachieve an efficient scaling with the system size, block sparse tensor trains -instances of tensor networks applied to function dictionaries - are used andthe self similarity of the problem is exploited. For the latter, we propose anapproach of gauge mediated weight sharing, inspired by notions of machinelearning, which significantly improves performance over previous approaches.The practical performance of the method is demonstrated numerically on threeone-dimensional systems - the Fermi-Pasta-Ulam-Tsingou system, rotatingmagnetic dipoles and classical particles interacting via modified Lennard-Jonespotentials. We highlight the ability of the method to recover these systems,requiring 1400 samples to recover the 50 particle Fermi-Pasta-Ulam-Tsingousystem to residuum of $5\times10^{-7}$, 900 samples to recover the 50 particlemagnetic dipole chain to residuum of $1.5\times10^{-4}$ and 7000 samples torecover the Lennard-Jones system of 10 particles to residuum$1.5\times10^{-2}$. The robustness against additive Gaussian noise isdemonstrated for the magnetic dipole system.

Block sparsity and gauge mediated weight sharing for learning dynamical laws from data