让数据说话：用于反问题的数据正则化算子学习理论
Let data talk: data-regularized operator learning theory for inverse problems
AI总结：
本文提出数据正则化算子学习（DaROL）方法解决PDE反问题，通过Tikhonov变分和贝叶斯推断等技术对数据进行正则化，将正则化与神经网络训练分离，证明了其等价于正则化逆映射的监督学习，并给出了平滑性条件和学习误差估计。
摘要：
正则化在将先验信息融入反问题中起着关键作用。虽然已经提出了许多深度学习方法来解决反问题，但确定在何处应用正则化仍然是一个关键的考量。典型的方法通过架构对神经网络进行正则化，其中神经网络函数对感兴趣的参数或正则化项进行参数化。我们引入了一种新颖的方法，称为“数据正则化算子学习”（DaROL）方法，旨在解决 PDE 反问题。DaROL 方法在数据上训练神经网络，并通过 Tikhonov 变分方法和贝叶斯推断等常用技术进行正则化。DaROL 方法在不同框架之间提供了灵活性，更快的反问题求解速度，以及将正则化和神经网络训练分离的更简单结构。我们证明了在正则化数据上训练神经网络等价于正则化逆映射的监督学习。此外，我们提供了这种正则化逆映射平滑性的充分条件，并估计了关于神经网络大小和训练样本数量的学习误差。
英文摘要：
Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial consideration. Typical methods regularize neural networks via architecture, wherein neural network functions parametrize the parameter of interest or the regularization term. We introduce a novel approach, denoted as the "data-regularized operator learning" (DaROL) method, designed to address PDE inverse problems. The DaROL method trains a neural network on data, regularized through common techniques such as Tikhonov variational methods and Bayesian inference. The DaROL method offers flexibility across different frameworks, faster inverse problem-solving, and a simpler structure that separates regularization and neural network training. We demonstrate that training a neural network on the regularized data is equivalent to supervised learning for a regularized inverse map. Furthermore, we provide sufficient conditions for the smoothness of such a regularized inverse map and estimate the learning error in terms of neural network size and the number of training samples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
