Multilevel Architectures and Algorithms in Deep Learning

深度神经网络（dnn）的设计及其训练是机器学习的核心问题。这些领域的进步是这些技术取得成功的动力之一。然而，在学习过程中，为了找到合适的网络结构和相应的超参数，以获得期望的DNN行为，往往仍然需要繁琐的实验和人工交互。拟议项目的战略目标是提供算法手段来改善这种情况。我们有条不紊的方法依赖于完善的数学技术：确定基本算法量并为它们构建后验估计，确定并始终如一地利用给定问题类的适当拓扑框架，为dnn建立多层结构，以解释dnn仅实现与输入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，我们将建立一类新的深度学习自适应多层算法，它不仅优化固定的DNN，而且在优化循环中自适应地改进和扩展DNN架构。这个概念并不局限于特定的网络架构，我们将研究前馈神经网络、ResNets和pin作为相关的例子。因此，我们的集成方法将能够通过基于后验估计的算法策略取代许多当前的手动调优技术。此外，与人工设计的DNN相比，我们的算法将减少训练的计算工作量和结果DNN的大小，使深度学习的使用在许多方面更有效。最后，从长远来看，我们的算法方法有可能提高训练后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.