National University of Computer and Emerging Sciences

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.