Learning on big data brings success for artificial intelligence (AI), but the annotation and training costs are expensive. In future, learning on small data that approximates the generalization ability of big data is one of the ultimate purposes of AI, which requires machines to recognize objectives and scenarios relying on small data as humans. A series of learning topics is going on this way such as active learning and few-shot learning. However, there are few theoretical guarantees for their generalization performance. Moreover, most of their settings are passive, that is, the label distribution is explicitly controlled by finite training resources from known distributions. This survey follows the agnostic active sampling theory under a PAC (Probably Approximately Correct) framework to analyze the generalization error and label complexity of learning on small data in model-agnostic supervised and unsupervised fashion. Considering multiple learning communities could produce small data representation and related topics have been well surveyed, we thus subjoin novel geometric representation perspectives for small data: the Euclidean and non-Euclidean (hyperbolic) mean, where the optimization solutions including the Euclidean gradients, non-Euclidean gradients, and Stein gradient are presented and discussed. Later, multiple learning communities that may be improved by learning on small data are summarized, which yield data-efficient representations, such as transfer learning, contrastive learning, graph representation learning. Meanwhile, we find that the meta-learning may provide effective parameter update policies for learning on small data. Then, we explore multiple challenging scenarios for small data, such as the weak supervision and multi-label. Finally, multiple data applications that may benefit from efficient small data representation are surveyed.

Learning from noisy data has attracted much attention, where most methods focus on closed-set label noise. However, a more common scenario in the real world is the presence of both open-set and closed-set noise. Existing methods typically identify and handle these two types of label noise separately by designing a specific strategy for each type. However, in many real-world scenarios, it would be challenging to identify open-set examples, especially when the dataset has been severely corrupted. Unlike the previous works, we explore how models behave when faced open-set examples, and find that a part of open-set examples gradually get integrated into certain known classes, which is beneficial for the seperation among known classes. Motivated by the phenomenon, in this paper, we propose a novel two-step contrastive learning method called CECL, which aims to deal with both types of label noise by exploiting the useful information of open-set examples. Specifically, we incorporate some open-set examples into closed-set classes to enhance performance while treating others as delimiters to improve representative ability. Extensive experiments on synthetic and real-world datasets with diverse label noise demonstrate that CECL can outperform state-of-the-art methods.

Although physics-informed neural networks(PINNs) have progressed a lot in many real applications recently, there remains problems to be further studied, such as achieving more accurate results, taking less training time, and quantifying the uncertainty of the predicted results. Recent advances in PINNs have indeed significantly improved the performance of PINNs in many aspects, but few have considered the effect of variance in the training process. In this work, we take into consideration the effect of variance and propose our VI-PINNs to give better predictions. We output two values in the final layer of the network to represent the predicted mean and variance respectively, and the latter is used to represent the uncertainty of the output. A modified negative log-likelihood loss and an auxiliary task are introduced for fast and accurate training. We perform several experiments on a wide range of different problems to highlight the advantages of our approach. The results convey that our method not only gives more accurate predictions but also converges faster.