We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator Am depending non-linearly on a parameter m and operating on a function u. In the inversion step both u and m are unknown but we are only interested in recovering m. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for Am, we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.

翻译：我们建立了一类反问题的Lipschitz稳定性性质。在该类问题中，相关的正问题由积分算子Am描述，该算子非线性地依赖于参数m并作用于函数u。在反演过程中，u和m均未知，但我们仅关注恢复m。我们讨论了弹性力学方程（在地震学中的应用）以及电磁理论中逆散射问题的这类反问题实例。假设Am满足若干单射性和正则性条件，我们证明具有有限数据点的反问题是可解的，且解关于数据具有Lipschitz稳定性。我们展示了一个利用神经网络的重建实例。