一致凸神经网络与非稳态迭代网络Tikhonov（iNETT）方法
Uniformly convex neural networks and non-stationary iterated network Tikhonov (iNETT) method
AI总结：
针对不适定反问题，提出非稳态迭代网络Tikhonov（iNETT）方法，利用一致凸神经网络构建数据驱动的正则化器以避免估计正则化参数，并提供了严格的理论、算法及二维计算机断层扫描应用验证。
摘要：
我们提出了一种用于求解不适定反问题的非稳态迭代网络Tikhonov（iNETT）方法。iNETT采用深度神经网络构建数据驱动的正则化器，从而避免了估计最优正则化参数的困难任务。为实现iNETT的理论收敛性，我们引入了一致凸神经网络来构建数据驱动的正则化器。针对凸和一致凸神经网络的构建，我们提出了严格的理论和详细的算法。特别地，给定一般的神经网络架构，我们规定了充分的条件以实现一个分量凸或一致凸的训练好的神经网络；此外，我们提供了在现代U-net架构中实现凸性和一致凸性的具体示例。借助凸和一致凸神经网络工具，我们开发了iNETT算法并提供了严格的收敛性分析。最后，我们展示了iNETT算法在二维计算机断层扫描中的应用，其中数值算例说明了所提算法的有效性。
英文摘要：
We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating the optimal regularization parameter. To achieve the theoretical convergence of iNETT, we introduce uniformly convex neural networks to build the data-driven regularizer. Rigorous theories and detailed algorithms are proposed for the construction of convex and uniformly convex neural networks. In particular, given a general neural network architecture, we prescribe sufficient conditions to achieve a trained neural network which is component-wise convex or uniformly convex; moreover, we provide concrete examples of realizing convexity and uniform convexity in the modern U-net architecture. With the tools of convex and uniformly convex neural networks, the iNETT algorithm is developed and a rigorous convergence analysis is provided. Lastly, we show applications of the iNETT algorithm in 2D computerized tomography, where numerical examples illustrate the efficacy of the proposed algorithm.
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