Multilevel Architectures and Algorithms in Deep Learning

深度神经网络(DNN)的设计和训练是机器学习的核心问题。这些领域的进步是这些技术成功的动力之一。然而，在学习过程中，仍然需要进行繁琐的实验和人工交互，以找到合适的网络结构和相应的超参数来获得DNN的期望行为。拟议项目的战略目标是提供算法手段来改善这种情况。我们的方法依赖于成熟的数学技术：识别基本算法量并构造它们的后验估计，识别并一致地利用给定问题类的合适的拓扑框架，建立DNN的多级结构来解释DNN只实现与输入和输出数据相关的连续非线性映射的离散近似这一事实。将这一思想与新的算法控制策略和预处理相结合，建立了一类新的用于深度学习的自适应多层算法，它不仅可以优化固定的DNN，还可以在优化循环期间自适应地细化和扩展DNN结构。这个概念并不局限于特定的网络结构，我们将研究前馈神经网络、ResNet和PINN作为相关的例子。因此，我们的综合方法将能够用基于后验估计的算法策略来取代许多当前的手动调整技术。此外，与人工设计的算法相比，我们的算法将减少训练的计算量，并减少生成的DNN的大小，从而在许多方面使深度学习的使用更加有效。最后，从长远来看，我们的算法方法有可能提高所得到的训练DNN的可靠性和可解释性。

The design of deep neural networks (DNNs) and their training is a central issue in machine learning. Progress in these areas is one of the driving forces for the success of these technologies. Nevertheless, tedious experimentation and human interaction is often still needed during the learning process to find an appropriate network structure and corresponding hyperparameters to obtain the desired behavior of a DNN. The strategic goal of the proposed project is to provide algorithmic means to improve this situation. Our methodical approach relies on well established mathematical techniques: identify fundamental algorithmic quantities and construct a-posteriori estimates for them, identify and consistently exploit an appropriate topological framework for the given problem class, establish a multilevel structure for DNNs to account for the fact that DNNs only realize a discrete approximation of a continuous nonlinear mapping relating input to output data. Combining this idea with novel algorithmic control strategies and preconditioning, we will establish the new class of adaptive multilevel algorithms for deep learning, which not only optimize a fixed DNN, but also adaptively refine and extend the DNN architecture during the optimization loop. This concept is not restricted to a particular network architecture, and we will study feedforward neural networks, ResNets, and PINNs as relevant examples. Our integrated approach will thus be able to replace many of the current manual tuning techniques by algorithmic strategies, based on a-posteriori estimates. Moreover, our algorithm will reduce the computational effort for training and also the size of the resulting DNN, compared to a manually designed counterpart, making the use of deep learning more efficient in many aspects. Finally, in the long run our algorithmic approach has the potential to enhance the reliability and interpretability of the resulting trained DNN.