We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. We assume that the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.

翻译：我们研究在仅能获得含噪测量数据时反问题解的构建。假设该问题可用非连续可逆的无穷维正向算子建模，随后将该正向算子限制于有限维空间，使逆算子满足Lipschitz连续性。针对该逆算子，我们证明存在一种神经网络能鲁棒逼近该算子。此外，我们表明此类神经网络可通过适当扰动的训练数据学习获得。我们论证了该方法对实际应用广泛反问题的适用性，并给出支持理论结果的数值示例。