论文标题
深度神经网络可以稳定解决高维，嘈杂的非线性反问题
Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems
论文作者
论文摘要
我们研究只有可用的嘈杂测量时，重建反问题解决方案的问题。我们假设问题可以用无限的远期操作员建模，该操作员并非不断可逆。然后，我们将该前向操作员限制为有限维空间，以使逆向Lipschitz连续。对于逆操作员，我们证明存在一个神经网络，该网络是操作员的健壮到噪声近似。此外，我们表明可以从适当的干扰培训数据中学到这些神经网络。我们证明了这种方法对实践感兴趣的各种反相反问题的可接受性。给出了支持理论发现的数值示例。
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.
