分散数据的数值插值旨在根据某些观察到的点估算目标点的值。传统方法通过构建结合多个基础函数的插值函数来产生估计。这些方法要求明确定义基础功能，从而在实际情况下极大地限制了其应用。最近的进步表现出一种替代策略，该策略可以直接使用机器学习技术（例如深度神经网络）从观察到的点学习插值功能。该策略虽然很有希望，但不能有效利用观察到的点和目标点之间的相关性，因为它可以分别处理这些类型的点。在这里，我们提出了一种基于学习的方法，使用变压器的编码器表示（因此称为NIERT）。 Niert将每个目标点的值视为蒙版令牌，它可以以统一的方式处理目标点并观察到点。通过计算目标点和观察点之间的部分自我注意，NIERT获得了利用这些点之间相关性的优势，更重要的是，避免了目标点在观察到的点上意外干扰。 NIERT还使用预训练技术进一步提高其准确性。在三个代表性数据集上，包括两个合成数据集和一个现实世界数据集，Niert优于现有方法，例如，在用于温度字段重建的TFRD-ADLET数据集上，Niert达到了$ 1.897 \ times 10^{ - 3} $ $ 1.897 ，比基于变压器的方法要好得多（MAE：$ 27.074 \ times 10^{ - 3} $）。这些结果清楚地表明了NIERT的准确性及其在多个实际领域中应用的潜力。

虽然外源变量对时间序列分析的性能改善有重大影响，但在当前的连续方法中很少考虑这些序列间相关性和时间依赖性。多元时间序列的动力系统可以用复杂的未知偏微分方程（PDE）进行建模，这些方程（PDE）在科学和工程的许多学科中都起着重要作用。在本文中，我们提出了一个任意步骤预测的连续时间模型，以学习多元时间序列中的未知PDE系统，其管理方程是通过自我注意和封闭的复发神经网络参数化的。所提出的模型\下划线{变量及其对目标系列的影响。重要的是，使用特殊设计的正则化指南可以将模型简化为正则化的普通微分方程（ODE）问题，这使得可以触犯的PDE问题以获得数值解决方案，并且可行，以预测目标序列的多个未来值。广泛的实验表明，我们提出的模型可以在强大的基准中实现竞争精度：平均而言，它通过降低RMSE的$ 9.85 \％$和MAE的MAE $ 13.98 \％$的基线表现优于最佳基准，以获得任意步骤预测的MAE $。

物理信息的神经网络（PINN）是神经网络（NNS），它们作为神经网络本身的组成部分编码模型方程，例如部分微分方程（PDE）。如今，PINN是用于求解PDE，分数方程，积分分化方程和随机PDE的。这种新颖的方法已成为一个多任务学习框架，在该框架中，NN必须在减少PDE残差的同时拟合观察到的数据。本文对PINNS的文献进行了全面的综述：虽然该研究的主要目标是表征这些网络及其相关的优势和缺点。该综述还试图将出版物纳入更广泛的基于搭配的物理知识的神经网络，这些神经网络构成了香草·皮恩（Vanilla Pinn）以及许多其他变体，例如物理受限的神经网络（PCNN），各种HP-VPINN，变量HP-VPINN，VPINN，VPINN，变体。和保守的Pinn（CPINN）。该研究表明，大多数研究都集中在通过不同的激活功能，梯度优化技术，神经网络结构和损耗功能结构来定制PINN。尽管使用PINN的应用范围广泛，但通过证明其在某些情况下比有限元方法（FEM）等经典数值技术更可行的能力，但仍有可能的进步，最著名的是尚未解决的理论问题。

时间序列数据在现实世界应用中无处不在。但是，最常见的问题之一是，时间序列数据可能会通过数据收集过程的固有性质丢失值。因此，必须从多元（相关）时间序列数据中推出缺失值，这对于改善预测性能的同时做出准确的数据驱动决策至关重要。插补的常规工作简单地删除缺失值或基于平均/零填充它们。尽管基于深层神经网络的最新作品显示出了显着的结果，但它们仍然有一个限制来捕获多元时间序列的复杂生成过程。在本文中，我们提出了一种用于多变量时间序列数据的新型插补方法，称为sting（使用GAN基于自我注意的时间序列插补网络）。我们利用生成的对抗网络和双向复发性神经网络来学习时间序列的潜在表示。此外，我们引入了一种新型的注意机制，以捕获整个序列的加权相关性，并避免无关序列带来的潜在偏见。三个现实世界数据集的实验结果表明，刺痛在插补精度以及具有估算值的下游任务方面优于现有的最新方法。

Many real-world applications require the prediction of long sequence time-series, such as electricity consumption planning. Long sequence time-series forecasting (LSTF) demands a high prediction capacity of the model, which is the ability to capture precise long-range dependency coupling between output and input efficiently. Recent studies have shown the potential of Transformer to increase the prediction capacity. However, there are several severe issues with Transformer that prevent it from being directly applicable to LSTF, including quadratic time complexity, high memory usage, and inherent limitation of the encoder-decoder architecture. To address these issues, we design an efficient transformer-based model for LSTF, named Informer, with three distinctive characteristics: (i) a ProbSparse self-attention mechanism, which achieves O(L log L) in time complexity and memory usage, and has comparable performance on sequences' dependency alignment. (ii) the self-attention distilling highlights dominating attention by halving cascading layer input, and efficiently handles extreme long input sequences. (iii) the generative style decoder, while conceptually simple, predicts the long time-series sequences at one forward operation rather than a step-by-step way, which drastically improves the inference speed of long-sequence predictions. Extensive experiments on four large-scale datasets demonstrate that Informer significantly outperforms existing methods and provides a new solution to the LSTF problem.

通过深度学习（DL）大大扩展了数据驱动故障诊断模型的范围。然而，经典卷积和反复化结构具有计算效率和特征表示的缺陷，而基于注意机制的最新变压器架构尚未应用于该字段。为了解决这些问题，我们提出了一种新颖的时变电片（TFT）模型，其灵感来自序列加工的香草变压器大规模成功。特别是，我们设计了一个新的笨蛋和编码器模块，以从振动信号的时频表示（TFR）中提取有效抽象。在此基础上，本文提出了一种基于时变电片的新的端到端故障诊断框架。通过轴承实验数据集的案例研究，我们构建了最佳变压器结构并验证了其故障诊断性能。与基准模型和其他最先进的方法相比，证明了所提出的方法的优越性。

Remaining Useful Life (RUL) estimation plays a critical role in Prognostics and Health Management (PHM). Traditional machine health maintenance systems are often costly, requiring sufficient prior expertise, and are difficult to fit into highly complex and changing industrial scenarios. With the widespread deployment of sensors on industrial equipment, building the Industrial Internet of Things (IIoT) to interconnect these devices has become an inexorable trend in the development of the digital factory. Using the device's real-time operational data collected by IIoT to get the estimated RUL through the RUL prediction algorithm, the PHM system can develop proactive maintenance measures for the device, thus, reducing maintenance costs and decreasing failure times during operation. This paper carries out research into the remaining useful life prediction model for multi-sensor devices in the IIoT scenario. We investigated the mainstream RUL prediction models and summarized the basic steps of RUL prediction modeling in this scenario. On this basis, a data-driven approach for RUL estimation is proposed in this paper. It employs a Multi-Head Attention Mechanism to fuse the multi-dimensional time-series data output from multiple sensors, in which the attention on features is used to capture the interactions between features and attention on sequences is used to learn the weights of time steps. Then, the Long Short-Term Memory Network is applied to learn the features of time series. We evaluate the proposed model on two benchmark datasets (C-MAPSS and PHM08), and the results demonstrate that it outperforms the state-of-art models. Moreover, through the interpretability of the multi-head attention mechanism, the proposed model can provide a preliminary explanation of engine degradation. Therefore, this approach is promising for predictive maintenance in IIoT scenarios.

时间变化数量的估计是医疗保健和金融等领域决策的基本组成部分。但是，此类估计值的实际实用性受到它们量化预测不确定性的准确程度的限制。在这项工作中，我们解决了估计高维多元时间序列的联合预测分布的问题。我们提出了一种基于变压器体系结构的多功能方法，该方法使用基于注意力的解码器估算关节分布，该解码器可被学会模仿非参数Copulas的性质。最终的模型具有多种理想的属性：它可以扩展到数百个时间序列，支持预测和插值，可以处理不规则和不均匀的采样数据，并且可以在训练过程中无缝地适应丢失的数据。我们从经验上证明了这些属性，并表明我们的模型在多个现实世界数据集上产生了最新的预测。

Time series anomaly detection has applications in a wide range of research fields and applications, including manufacturing and healthcare. The presence of anomalies can indicate novel or unexpected events, such as production faults, system defects, or heart fluttering, and is therefore of particular interest. The large size and complex patterns of time series have led researchers to develop specialised deep learning models for detecting anomalous patterns. This survey focuses on providing structured and comprehensive state-of-the-art time series anomaly detection models through the use of deep learning. It providing a taxonomy based on the factors that divide anomaly detection models into different categories. Aside from describing the basic anomaly detection technique for each category, the advantages and limitations are also discussed. Furthermore, this study includes examples of deep anomaly detection in time series across various application domains in recent years. It finally summarises open issues in research and challenges faced while adopting deep anomaly detection models.

Multivariate time series forecasting (MTSF) is a fundamental problem in numerous real-world applications. Recently, Transformer has become the de facto solution for MTSF, especially for the long-term cases. However, except for the one forward operation, the basic configurations in existing MTSF Transformer architectures were barely carefully verified. In this study, we point out that the current tokenization strategy in MTSF Transformer architectures ignores the token uniformity inductive bias of Transformers. Therefore, the vanilla MTSF transformer struggles to capture details in time series and presents inferior performance. Based on this observation, we make a series of evolution on the basic architecture of the vanilla MTSF transformer. We vary the flawed tokenization strategy, along with the decoder structure and embeddings. Surprisingly, the evolved simple transformer architecture is highly effective, which successfully avoids the over-smoothing phenomena in the vanilla MTSF transformer, achieves a more detailed and accurate prediction, and even substantially outperforms the state-of-the-art Transformers that are well-designed for MTSF.

异步时间序列是一个多元时间序列，在该时间序列中，所有通道都被观察到异步独立的，使得时间序列在对齐时极为稀疏。我们经常在具有复杂的观察过程（例如医疗保健，气候科学和天文学）的应用中观察到这种影响，仅举几例。由于异步性质，它们对深度学习体系结构构成了重大挑战，假定给他们的时间序列定期采样，完全观察并与时间对齐。本文提出了一个新颖的框架，我们称深卷积集功能（DCSF），该功能高度可扩展且有效，对于异步时间序列分类任务。随着深度学习体系结构的最新进展，我们引入了一个模型，该模型不变了，在此订单中呈现了时间序列的频道。我们探索卷积神经网络，该网络对定期采样和完全观察到的时间序列的紧密相关的问题分类进行了很好的研究，以编码设置元素。我们评估DCSF的ASTS分类和在线（每个时间点）ASTS分类。我们在多个现实世界和合成数据集上进行的广泛实验验证了建议的模型在准确性和运行时间方面的表现优于一系列最新模型。

Attention-based neural networks, such as Transformers, have become ubiquitous in numerous applications, including computer vision, natural language processing, and time-series analysis. In all kinds of attention networks, the attention maps are crucial as they encode semantic dependencies between input tokens. However, most existing attention networks perform modeling or reasoning based on representations, wherein the attention maps of different layers are learned separately without explicit interactions. In this paper, we propose a novel and generic evolving attention mechanism, which directly models the evolution of inter-token relationships through a chain of residual convolutional modules. The major motivations are twofold. On the one hand, the attention maps in different layers share transferable knowledge, thus adding a residual connection can facilitate the information flow of inter-token relationships across layers. On the other hand, there is naturally an evolutionary trend among attention maps at different abstraction levels, so it is beneficial to exploit a dedicated convolution-based module to capture this process. Equipped with the proposed mechanism, the convolution-enhanced evolving attention networks achieve superior performance in various applications, including time-series representation, natural language understanding, machine translation, and image classification. Especially on time-series representation tasks, Evolving Attention-enhanced Dilated Convolutional (EA-DC-) Transformer outperforms state-of-the-art models significantly, achieving an average of 17% improvement compared to the best SOTA. To the best of our knowledge, this is the first work that explicitly models the layer-wise evolution of attention maps. Our implementation is available at https://github.com/pkuyym/EvolvingAttention

各种深度学习模型，尤其是一些最新的基于变压器的方法，已大大改善了长期时间序列预测的最新性能。但是，这些基于变压器的模型遭受了严重的恶化性能，并延长了输入长度除了使用扩展的历史信息。此外，这些方法倾向于在长期预测中处理复杂的示例，并增加模型复杂性，这通常会导致计算的显着增加和性能较低的鲁棒性（例如，过度拟合）。我们提出了一种新型的神经网络架构，称为Treedrnet，以进行更有效的长期预测。受稳健回归的启发，我们引入了双重残差链接结构，以使预测更加稳健。对Kolmogorov-Arnold表示定理进行了明确的介绍，并明确介绍了特征选择，模型集合和树结构，以进一步利用扩展输入序列，从而提高了可靠的输入序列和Treedrnet的代表力。与以前的顺序预测工作的深层模型不同，Treedrnet完全建立在多层感知下，因此具有很高的计算效率。我们广泛的实证研究表明，Treedrnet比最先进的方法更有效，将预测错误降低了20％至40％。特别是，Treedrnet的效率比基于变压器的方法高10倍。该代码将很快发布。

Natural laws are often described through differential equations yet finding a differential equation that describes the governing law underlying observed data is a challenging and still mostly manual task. In this paper we make a step towards the automation of this process: we propose a transformer-based sequence-to-sequence model that recovers scalar autonomous ordinary differential equations (ODEs) in symbolic form from time-series data of a single observed solution of the ODE. Our method is efficiently scalable: after one-time pretraining on a large set of ODEs, we can infer the governing laws of a new observed solution in a few forward passes of the model. Then we show that our model performs better or on par with existing methods in various test cases in terms of accurate symbolic recovery of the ODE, especially for more complex expressions.

像长期短期内存网络（LSTMS）和门控复发单元（GRUS）相同的经常性神经网络（RNN）是建模顺序数据的流行选择。它们的门控机构允许以来自传入观测的新信息在隐藏状态中编码的先前历史。在许多应用程序中，例如医疗记录，观察时间是不规则的并且携带重要信息。然而，LSTM和GRUS在观察之间假设恒定的时间间隔。为了解决这一挑战，我们提出了连续的经常性单位（CRU）-A神经结构，可以自然地处理观察之间的不规则时间间隔。 CRU的浇注机制采用卡尔曼滤波器的连续制剂，并且根据线性随机微分方程（SDE）和（2）潜伏状态在新观察进入时，在（1）之间的连续潜在传播之间的交替。在实证研究，我们表明CRU可以比神经常规差分方程（神经颂歌）的模型更好地插值不规则时间序列。我们还表明，我们的模型可以从IM-AGES推断动力学，并且卡尔曼有效地单挑出候选人的候选人，从而从嘈杂的观察中获得有价值的状态更新。

最近，对于长期时间序列预测（LTSF）任务，基于变压器的解决方案激增。尽管过去几年的表现正在增长，但我们质疑这项研究中这一研究的有效性。具体而言，可以说，变形金刚是最成功的解决方案，是在长序列中提取元素之间的语义相关性。但是，在时间序列建模中，我们要在一组连续点的有序集中提取时间关系。在采用位置编码和使用令牌将子系列嵌入变压器中的同时，有助于保留某些订购信息，但\ emph {置换不变}的自我注意力专注机制的性质不可避免地会导致时间信息损失。为了验证我们的主张，我们介绍了一组名为LTSF线性的令人尴尬的简单单层线性模型，以进行比较。在九个现实生活数据集上的实验结果表明，LTSF线性在所有情况下都超过现有的基于变压器的LTSF模型，并且通常要大幅度较大。此外，我们进行了全面的经验研究，以探索LTSF模型各种设计元素对其时间关系提取能力的影响。我们希望这一令人惊讶的发现为LTSF任务打开了新的研究方向。我们还主张重新审视基于变压器解决方案对其他时间序列分析任务（例如，异常检测）的有效性。代码可在：\ url {https://github.com/cure-lab/ltsf-linear}中获得。

多元时间序列（MTS）预测在广泛的应用中起着至关重要的作用。最近，时空图神经网络（STGNN）已成为越来越流行的MTS预测方法。 STGNN通过图神经网络和顺序模型共同对MTS的空间和时间模式进行建模，从而显着提高了预测准确性。但是受模型复杂性的限制，大多数STGNN仅考虑短期历史MTS数据，例如过去一个小时的数据。但是，需要根据长期的历史MTS数据来分析时间序列的模式及其之间的依赖关系（即时间和空间模式）。为了解决这个问题，我们提出了一个新颖的框架，其中STGNN通过可扩展的时间序列预训练模型（步骤）增强。具体而言，我们设计了一个预训练模型，以从非常长期的历史时间序列（例如，过去两周）中有效地学习时间模式并生成细分级表示。这些表示为短期时间序列输入到STGNN提供了上下文信息，并促进了时间序列之间的建模依赖关系。三个公共现实世界数据集的实验表明，我们的框架能够显着增强下游STGNN，并且我们的训练前模型可恰当地捕获时间模式。

Ordinary Differential Equations (ODE)-based models have become popular foundation models to solve many time-series problems. Combining neural ODEs with traditional RNN models has provided the best representation for irregular time series. However, ODE-based models require the trajectory of hidden states to be defined based on the initial observed value or the last available observation. This fact raises questions about how long the generated hidden state is sufficient and whether it is effective when long sequences are used instead of the typically used shorter sequences. In this article, we introduce CrossPyramid, a novel ODE-based model that aims to enhance the generalizability of sequences representation. CrossPyramid does not rely only on the hidden state from the last observed value; it also considers ODE latent representations learned from other samples. The main idea of our proposed model is to define the hidden state for the unobserved values based on the non-linear correlation between samples. Accordingly, CrossPyramid is built with three distinctive parts: (1) ODE Auto-Encoder to learn the best data representation. (2) Pyramidal attention method to categorize the learned representations (hidden state) based on the relationship characteristics between samples. (3) Cross-level ODE-RNN to integrate the previously learned information and provide the final latent state for each sample. Through extensive experiments on partially-observed synthetic and real-world datasets, we show that the proposed architecture can effectively model the long gaps in intermittent series and outperforms state-of-the-art approaches. The results show an average improvement of 10\% on univariate and multivariate datasets for both forecasting and classification tasks.

来自数据的顺序模式是各种时间序列预测任务的核心。深度学习模型大大优于许多传统模型，但是这些黑框模型通常缺乏预测和决策的解释性。为了揭示具有可理解的数学表达式的潜在趋势，科学家和经济学家倾向于使用部分微分方程（PDE）来解释顺序模式的高度非线性动力学。但是，它通常需要领域专家知识和一系列简化的假设，这些假设并不总是实用的，并且可能偏离不断变化的世界。是否可以动态地学习与数据的差异关系以解释时间不断发展的动态？在这项工作中，我们提出了一个学习框架，该框架可以自动从顺序数据中获取可解释的PDE模型。特别是，该框架由可学习的差分块组成，称为$ p $ blocks，事实证明，该框架能够近似于理论上随着时间不断变化的复杂连续功能。此外，为了捕获动力学变化，该框架引入了元学习控制器，以动态优化混合PDE模型的超参数。 《时代》系列预测金融，工程和健康数据的广泛实验表明，我们的模型可以提供有价值的解释性并实现与最先进模型相当的性能。从经验研究中，我们发现学习一些差异操作员可能会捕获无需大量计算复杂性的顺序动力学的主要趋势。

Future surveys such as the Legacy Survey of Space and Time (LSST) of the Vera C. Rubin Observatory will observe an order of magnitude more astrophysical transient events than any previous survey before. With this deluge of photometric data, it will be impossible for all such events to be classified by humans alone. Recent efforts have sought to leverage machine learning methods to tackle the challenge of astronomical transient classification, with ever improving success. Transformers are a recently developed deep learning architecture, first proposed for natural language processing, that have shown a great deal of recent success. In this work we develop a new transformer architecture, which uses multi-head self attention at its core, for general multi-variate time-series data. Furthermore, the proposed time-series transformer architecture supports the inclusion of an arbitrary number of additional features, while also offering interpretability. We apply the time-series transformer to the task of photometric classification, minimising the reliance of expert domain knowledge for feature selection, while achieving results comparable to state-of-the-art photometric classification methods. We achieve a logarithmic-loss of 0.507 on imbalanced data in a representative setting using data from the Photometric LSST Astronomical Time-Series Classification Challenge (PLAsTiCC). Moreover, we achieve a micro-averaged receiver operating characteristic area under curve of 0.98 and micro-averaged precision-recall area under curve of 0.87.