Many problems in science and engineering can be mathematically modeled using partial differential equations (PDEs), which are essential for fields like computational fluid dynamics (CFD), molecular dynamics, and dynamical systems. Although traditional numerical methods like the finite difference/element method are widely used, their computational inefficiency, due to the large number of iterations required, has long been a challenge. Recently, deep learning (DL) has emerged as a promising alternative for solving PDEs, offering new paradigms beyond conventional methods. Despite the growing interest in techniques like physics-informed neural networks (PINNs), a systematic review of the diverse neural network (NN) approaches for PDEs is still missing. This survey fills that gap by categorizing and reviewing the current progress of deep NNs (DNNs) for PDEs. Unlike previous reviews focused on specific methods like PINNs, we offer a broader taxonomy and analyze applications across scientific, engineering, and medical fields. We also provide a historical overview, key challenges, and future trends, aiming to serve both researchers and practitioners with insights into how DNNs can be effectively applied to solve PDEs.

In this article, we propose a structure-aligned generative adversarial network framework to improve zero-shot learning (ZSL) by mitigating the semantic gap, domain shift, and hubness problem. The proposed framework contains two parts, i.e., a generative adversarial network with a softmax classifier part, and a structure-aligned part. In the first part, the generative adversarial network aims at generating pseudovisual features through the guiding generator and discriminator play the minimax two-player game together. At the same time, the softmax classifier is committed to increasing the interclass distance and reducing intraclass distance. Then, the harmful effect of domain shift and hubness problems can be mitigated. In another part, we introduce a structure-aligned module where the structural consistency between visual space and semantic space is learned. By aligning the structure between visual space and semantic space, the semantic gap between them can be bridged. The performance of classification is improved when the structure-aligned visual-semantic embedding space is transferred to the unseen classes. Our framework reformulates the ZSL as a standard fully supervised classification task using the pseudovisual features of unseen classes. Extensive experiments conducted on five benchmark data sets demonstrate that the proposed framework significantly outperforms state-of-the-art methods in both conventional and generalized settings.

In recent years, the signal diagnosis on device operating under industrial environment has attracted increasingly attention. Most data-driven signal diagnosis methods are based on closed set assumption that class sets of training and test data are consistent. However, in industrial scenarios, during the running process of the device, the operating environment and condition may change over time, continuing generating data belong to unknown classes with new characteristics and distribution. The unknown classes usually reflect new modes or faults of the device need to be captured. They are unavailable in training phase, contradicting the closed set assumption. Existing methods are inappropriate to this type of open set classification requiring to classify known classes and recognize unknown classes. To address this challenging problem, this paper proposes a generic open set signal classification method. First, we apply Fourier transform to convert the sensor signals from time domain to frequency domain, then data in the time and frequency domains are fused. Next, a variational encoder-classifier network is proposed to classify known classes and learn the distribution of feature space to extract robust latent features. Finally, based on extreme value theory and entropy, a pair of discriminators determine whether samples belong to unknown or not. The experimental results on two vibration signal datasets from bearings and nuclear reactor demonstrate the effectiveness and superiority of our proposed open set signal classification method, especially in practical applications.

Existing object detection models are mainly trained on large-scale labeled datasets. However, annotating data for novel aerial object classes is expensive since it is time-consuming and may require expert knowledge. Thus, it is desirable to study label-efficient object detection methods on aerial images. In this work, we propose a zero-shot method for aerial object detection named visual Description Regularization, or DescReg. Concretely, we identify the weak semantic-visual correlation of the aerial objects and aim to address the challenge with prior descriptions of their visual appearance. Instead of directly encoding the descriptions into class embedding space which suffers from the representation gap problem, we propose to infuse the prior inter-class visual similarity conveyed in the descriptions into the embedding learning. The infusion process is accomplished with a newly designed similarity-aware triplet loss which incorporates structured regularization on the representation space. We conduct extensive experiments with three challenging aerial object detection datasets, including DIOR, xView, and DOTA. The results demonstrate that DescReg significantly outperforms the state-of-the-art ZSD methods with complex projection designs and generative frameworks, e.g., DescReg outperforms best reported ZSD method on DIOR by 4.5 mAP on unseen classes and 8.1 in HM. We further show the generalizability of DescReg by integrating it into generative ZSD methods as well as varying the detection architecture. Codes will be released at https://github.com/zq-zang/DescReg.