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
Synopsis
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.
Interpretation
Neural solvers for damped elastic waves no longer fit the raw partial differential equation directly; the equation is first reduced to a form in one similarity variable and the network fits only the reduced unknown. A standard physics-informed network writes only the residual and boundary conditions into its loss, which on time-dependent nonlinear systems brings gradient imbalance, stiffness and slow convergence. Across four cases the profiles, surfaces and contours stay closer to the analytical solutions of the reduced equations, and on the generalized damped symmetry case the total wall-clock time including derivation and training gives a speed-up of 1.53.
The network learns the unknown function in the similarity variable, and the learned result is reconstructed back to the original space and time variables through the invariant map. Lie symmetry analysis supplies those variables: it finds the variable combinations that leave the equation form unchanged, reduces the viscously damped equation to an ordinary differential equation, and writes the invariant condition as an extra loss term; causal weighting then cuts the time axis into ordered subdomains and weights later residuals by the loss already accumulated at earlier ones, so early times are learned first. All four cases share one fully connected network with four hidden layers of 128 neurons and an L-BFGS optimizer, with reference solutions obtained directly from the reduced equations.
Perspective
The demonstrated scope is unidirectional longitudinal motion in a nonlinear elastic wave equation with a small viscous damping term, together with the four invariant ordinary differential equations that Lie symmetry analysis yields for it. The reference is the analytical solution of those reduced equations, and the compared quantities are solution profiles, surfaces, contours, loss components, gradient norms and wall-clock time; the stated application areas are computational mechanics, material modeling, nondestructive testing and biomedical ultrasound. The proposed next step is to extend the approach to two- and three-dimensional elastodynamic models, more realistic viscoelastic or anisotropic materials, and inverse problems for damping coefficients and elastic parameters.
The wall-clock figures are stated as representative estimates from one computational environment, used to compare the preparation-training trade-off. The S-PINN preparation stage additionally derives infinitesimal generators, similarity variables, invariant reconstruction maps and reduced equations, and that cost is paid only once for a given equation and symmetry class.