2026/07/23 by Yuxiang He, Shuai Lu
#math.AP
This paper investigates recovery of an unknown coefficient in a semilinear wave equation defined on a bounded, open, and strictly convex domain in \(ℝ1+n\) with \(n ≥ 2\). We demonstrate that the unknown coefficient \(q\) appearing in the semilinear wave equation \(\square u + q um = 0\) with Neumann boundary conditions can be reconstructed with Hölder stability from the linearized Neumann-to-Dirichlet (NtD) map. Our approach combines first-order linearization with the Principle of Inclusion-Exclusion (PIE) identity, and employs geometric optics solutions for wave equations in two distinct regimes: the case \(m=2\) with \(q = q(x)\), and the case \(m ≥ 3\) with \(q = q(t,x)\). Furthermore, numerical examples illustrate that the unknown coefficient can also be effectively reconstructed using a neural network-based inversion algorithm within the framework of the least squares method.