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.

翻译：本文关注一类正向算子涉及未知函数的非线性逆问题。该问题存在于多种应用场景中，而未知函数的存在使其具有病态性，极具挑战性。此外，问题的非线性特征使得传统方法难以适用，现有研究通常通过线性化或假设已知未知函数来处理问题的简化版本。本文提出采用自监督学习直接求解涉及未知函数的非线性逆问题。我们特别聚焦于光声层析成像（PAT）中推导出的逆问题——这是一种兼具高分辨率与高对比度的混合医学成像技术。PAT可基于波动方程建模，测量数据提供了限制在曲面上的方程解与被测对象生物信息相关的初始压力。方程中的声波速度未知，我们的目标是同时确定初始压力与声波速度。在声速为初始压力函数的简单假设下，该问题转化为涉及未知函数的非线性逆问题。实验结果表明，所提出的框架能够成功运行。