Potential of data-driven discovery of nonlinear wave equations

Data-driven partial differential equation discovery techniques have been growing in importance in recent years. This paper discusses its possible applications in the field of finite-amplitude sound propagation and the related phenomena (such as generation of higher harmonics and shock formation). Two cases are investigated, namely the propagation of pressure pulses as travelling waves and the interference of pressure pulses, the former leading to the Westervelt equation and the latter to the Kuznetsov equation. It is shown how representative wave equations can be extracted from data obtained by numerically solving the compressible Navier-Stokes equations (up to the third order smallness in entropy changes) and subsequently employing the sparsity promoting regression techniques. The resulting error of the equation coefficients is about 7% in the nonlinear terms. The limitations of this approach are discussed (e.g., the appropriate capturing of thermal and viscous loss terms).