Learning rate is arguably the most important hyper-parameter to tune when training a neural network. As manually setting right learning rate remains a cumbersome process, adaptive learning rate algorithms aim at automating such a process. Motivated by the success of the Barzilai–Borwein (BB) step-size method in many gradient descent methods for solving convex problems, this paper aims at investigating the potential of the BB method for training neural networks. With strong motivation from related convergence analysis, the BB method is generalized to adaptive learning rate of mini-batch gradient descent. The experiments showed that, in contrast to many existing methods, the proposed BB method is highly insensitive to initial learning rate, especially in terms of generalization performance. Also, the BB method showed its advantages on both learning speed and generalization performance over other available methods.

Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data points and is a critical component for achieving high classification accuracy. Many existing deep learning approaches directly impose the fusion loss when training neural networks. In this work, inspired by the convection-diffusion ordinary differential equations (ODEs), we propose a novel diffusion residual network (Diff-ResNet), internally introduces diffusion into the architectures of neural networks. Under the structured data assumption, it is proved that the proposed diffusion block can increase the distance-diameter ratio that improves the separability of inter-class points and reduces the distance among local intra-class points. Moreover, this property can be easily adopted by the residual networks for constructing the separable hyperplanes. Extensive experiments of synthetic binary classification, semi-supervised graph node classification and few-shot image classification in various datasets validate the effectiveness of the proposed method.

The dynamic formulation of optimal transport has attracted growing interests in scientific computing and machine learning, and its computation requires to solve a PDE-constrained optimization problem. The classical Eulerian discretization based approaches suffer from the curse of dimensionality, which arises from the approximation of high-dimensional velocity field. In this work, we propose a deep learning based method to solve the dynamic optimal transport in high dimensional space. Our method contains three main ingredients: a carefully designed representation of the velocity field, the discretization of the PDE constraint along the characteristics, and the computation of high dimensional integral by Monte Carlo method in each time step. Specifically, in the representation of the velocity field, we apply the classical nodal basis function in time and the deep neural networks in space domain with the H1-norm regularization. This technique promotes the regularity of the velocity field in both time and space such that the discretization along the characteristic remains to be stable during the training process. Extensive numerical examples have been conducted to test the proposed method. Compared to other solvers of optimal transport, our method could give more accurate results in high dimensional cases and has very good scalability with respect to dimension. Finally, we extend our method to more complicated cases such as crowd motion problem.