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In this article, we are concerned with a nonlinear inverse problem with aforward operator involving an unknown function. The problem arises in diverseapplications and is challenging in the presence of an unknown function, whichmakes it ill-posed. Additionally, the nonlinear nature of the problem makes itdifficult to use traditional methods, and thus, the study addresses asimplified version of the problem by either linearizing it or assumingknowledge of the unknown function. Here, we propose self-supervised learning todirectly tackle a nonlinear inverse problem involving an unknown function. Inparticular, we focus on an inverse problem derived in photoacoustic tomograpy(PAT), which is a hybrid medical imaging with high resolution and contrast. PATcan be modeled based on the wave equation. The measured data provide thesolution to an equation restricted to surface and initial pressure of anequation that contains biological information on the object of interest. Thespeed of a sound wave in the equation is unknown. Our goal is to determine theinitial pressure and the speed of the sound wave simultaneously. Under a simpleassumption that sound speed is a function of the initial pressure, the problembecomes a nonlinear inverse problem involving an unknown function. Theexperimental results demonstrate that the proposed framework performssuccessfully.

Self-supervised learning for a nonlinear inverse problem with forward operator involving an unknown function arising in photoacoustic tomography