A Hybrid Physics-Informed Neural Network Framework for Computing Dispersion Relations of SH Waves in Generalized Hetrogeneous Layered Media with Applications

Abstract

This work presents a mathematical and computational framework for computing the dispersion relations of shear horizontal (SH) waves in continuously varying heterogeneous layered structures. The approach isolates the contribution of the heterogeneous layer from the complete dispersion relation, learns this contribution using a physics-informed neural network (PINN) and subsequently incorporates the trained model to determine the complete dispersion relation. The mathematical properties of the discretized problem are investigated, including the singularity of the finite-difference system and the oscillatory behavior of the layer solution, while a generalization-error estimate is established for the PINN approximation. The framework is first tested on a seismological configuration consisting of a heterogeneous sandstone layer over a granite half-space, where exponential heterogeneity is considered with independent variation rates in shear modulus and density. The proposed approach is validated against analytical solutions in special cases, while for general configurations the Haskell matrix method demonstrates convergence toward the continuously varying dispersion relation predicted by the PINN as the number of homogeneous sublayers increases. Parametric studies further confirm consistency with the underlying physics. An important feature of the method is that the heterogeneous-layer equation can be trained independently and reused in multiple settings. To demonstrate this, the same trained network is coupled with piezoelectric and piezomagnetic substrates governed by fundamentally different physical laws, highlighting the potential of the PINN framework as a reusable computational module for dispersion analysis in heterogeneous layered media.