  In this article, we are concerned with a nonlinear inverse problem with a
forward operator involving an unknown function. The problem arises in diverse
applications and is challenging in the presence of an unknown function, which
makes it ill-posed. Additionally, the nonlinear nature of the problem makes it
difficult to use traditional methods, and thus, the study addresses a
simplified version of the problem by either linearizing it or assuming
knowledge of the unknown function. Here, we propose self-supervised learning to
directly tackle a nonlinear inverse problem involving an unknown function. In
particular, we focus on an inverse problem derived in photoacoustic tomograpy
(PAT), which is a hybrid medical imaging with high resolution and contrast. PAT
can be modeled based on the wave equation. The measured data provide the
solution to an equation restricted to surface and initial pressure of an
equation that contains biological information on the object of interest. The
speed of a sound wave in the equation is unknown. Our goal is to determine the
initial pressure and the speed of the sound wave simultaneously. Under a simple
assumption that sound speed is a function of the initial pressure, the problem
becomes a nonlinear inverse problem involving an unknown function. The
experimental results demonstrate that the proposed framework performs
successfully.
