JostONet: A neural operator architecture for solving the Jost solution and scattering coefficients of the Schrödinger spectral problem

The Schrödinger spectral problem is a central topic in mathematical physics. In numerical inverse scattering transform (NIST), the reflection coefficient R(k) contained in the scattering data S must be repeatedly computed by solving the spectral problem at discrete wave number k. We propose a novel neural operator framework, the Jost operator network (JostONet), for fast inference of the Jost solution and its associated R(k), offering a promising alternative for computing R(k) in the NIST. JostONet is composed of three specialized modules: (i) High-energy region R_h: inspired by the asymptotic behavior of the Jost solution, a novel amplitude decomposition is derived, based on which a variable amplitude operator network is constructed. Normalization conditions and conservation property are embedded in the loss function, and hard boundary constraints are imposed. (ii) Intermediate-energy region R_m: The wave number k is treated as a degenerate functional variable, and a wave function operator network is constructed based on the multi-input operator network. (iii) Low-energy region R_l: a perturbation-wave function operator network is introduced, which exploits the perturbation expansion of the Jost solution with respect to k and is composed of a sequence of Deep Operator Networks. During training, a novel function space H˜^κ,η(R) is constructed based on Hermite polynomials to generate potential functions with Gaussian decay, which serve as inputs to the neural operators. JostONet achieves satisfactory predictive accuracy across all energy regions, with an inference speed at least an order of magnitude faster than traditional methods, and it is capable of generalizing to higher-order potentials in the space H˜^κ,η(R). In addition, we provide theoretical support and extensive numerical validation for the partitioning of k, along with detailed numerical analysis of each module.