A Physics-Informed Neural Network Based on the Separation of Variables for Solving a Distributed-Order Dispersive Generalized Hunter–Saxton Model with Fractional-Time Effects

Abstract

This paper presents and examines an extension of the generalized Hunter-Saxton equation to higher order of dispersion. The model in its classical form is defined as, . Where the dispersive/regularizing high-order term added to the model adds to the traveling-wave dynamics over the non-dispersive case. We measure the effects of the fractional-time by looking at time-fractional analogs of three standard operators the modified Riemann Liouville derivative, the -type derivative and the M-truncated derivative and show that the fractionalization changes the effective traveling-wave variable and introduces observable changes in the localization and propagation of waves. In order to unify retention of nonlocal memory effects, we also propose a distributed-order time-fractional correction where the time operator operates on the slope field , , Because Caputo-type distributed-order operators contain history integrals that cannot be directly addressed using standard automatic differentiation, we introduce a separation-of-variables physics-informed neural network (SVPINN) model that parametrizes the distributed-order correction by a trainable surrogate model. Lastly, numerical diagnostics of the related reduced dynamical system display high sensitivity to initial perturbations, limited transverse oscillations in some regimes, a non-isolated equilibrium manifold that causes switching between neutral drift, center-type oscillations, and saddle-type behavior. On the whole, it can be concluded that fourth-order dispersion and fractional-time processes and, to a great extent, distributed-order memory significantly add to the qualitative dynamics of Hunter Saxton-type model.