Conservation laws for Newtonian systems now come from data: symmetry conditions become linear equations on a lifted metric

Deriving conservation laws no longer starts from a known form of the equations of motion; it starts from phase-space trajectory data, learns the potential, and then finds symmetries on the extended geometry. Previously, determining Lie and Noether symmetries required solving symmetry partial differential equations that depend explicitly on the equations of motion, with the potential entering as an additional constraint. For the one-dimensional Ermakov-Pinney equation, the extended metric reconstructed from Gaussian-noisy mock trajectories is flat within reconstruction error, and the random-feature network returns three Killing vectors, matching the three expected for a two-dimensional flat space.

The procedure has four steps: a Hamiltonian network learns the potential from data, the potential defines the extended metric of the Riemannian or Lorentzian lift, a random-feature network solves the Killing and conformal Killing equations, and Noether's theorem yields the conserved quantities and their commutators. The random-feature network keeps its weights and biases fixed at random values and leaves only the expansion coefficients unknown, so the linear symmetry condition becomes a linear least-squares problem in those coefficients. For the one-dimensional harmonic oscillator, the Cotton-York tensor of the three-dimensional extended geometry from the Lorentzian lift averages close to zero, and the network reconstructs ten conformal Killing vectors whose commutators are independent of noise level and form the Lie algebra.

When a quartic perturbation is added to the harmonic oscillator, the broken symmetries survive as approximate symmetries, and how well they survive is controlled by the size of the perturbation. The unperturbed oscillator has a ten-dimensional conformal algebra under the Lorentzian lift, while analytically the quartic term leaves only the time-translation symmetry. For the smaller quartic correction, ten vector fields still fall below the residual threshold over one period with relative non-conservation below the stated level; for the larger correction only six vector fields satisfy that condition, and their relative non-conservation is larger.

The demonstrated scope is mock phase-space trajectories for the one-dimensional Ermakov-Pinney equation and the one-dimensional harmonic oscillator with and without a quartic perturbation, with Gaussian noise and medians over five independent networks; the random-feature network learns symmetries only within the region covered by the data, so the reconstructed conservation laws hold only there.

The curvature invariants of the extended geometry depend on second and third derivatives of the learned potential, which are more sensitive to reconstruction error than the potential itself, and the oscillation amplitude of the Cotton-York tensor grows with noise. Under the quartic perturbation the broken symmetries survive only as approximate symmetries whose residuals vary with perturbation strength, and the decision near the threshold depends on the chosen gap.