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# Lie symmetry analysis

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arXiv Nov 9, 2026
Conservation laws for Newtonian systems now come from data: symmetry conditions become linear equations on a lifted metric
Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.
Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.
Andronikos Paliathanasis（First author, Corresponding author）
Physics-Informed Neural Network Eisenhart-Duval lift Lie symmetry analysis
PLoS ONE Oct 27, 2026
Damped elastic waves no longer need a free search: Lie symmetry reduction plus causal weighting gives a 1.53× total speed-up on the generalized damped case
Neural solvers for the damped nonlinear elastic wave equation now train differently: Lie symmetry analysis reduces it to four ordinary differential equations in a single similarity variable, the network learns the reduced unknown, a symmetry loss constrains invariance, and residuals are weighted in time so early times are learned first. A standard physics-informed network relies only on the residual and boundary conditions, and on time-dependent nonlinear systems it tends to hit gradient imbalance and slow convergence. Across four cases, profiles, surfaces and contours stay closer to the analytical solutions of the reduced equations, and the generalized damped case shows a 1.53× total wall-clock speed-up including derivation and training.
Neural solvers for the damped nonlinear elastic wave equation now train differently: Lie symmetry analysis reduces it to four ordinary differential equations in a single similarity variable, the network learns the reduced unknown, a symmetry loss constrains invariance, and residuals are weighted in time so early times are learned first. A standard physics-informed network relies only on the residual and boundary conditions, and on time-dependent nonlinear systems it tends to hit gradient imbalance and slow convergence. Across four cases, profiles, surfaces and contours stay closer to the analytical solutions of the reduced equations, and the generalized damped case shows a 1.53× total wall-clock speed-up including derivation and training.
National University of Computer and Emerging Sciences University of Hafr Al-Batin
Muhammad Aamir Israr（First author）Muhammad Israr（First author）
S-PINN Physics-Informed Neural Network Lie symmetry analysis Causality-based temporal weighting Nonlinear damped elastodynamic PDEs

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