分析了无限维函数空间之间地图的深层替代物的近似速率，例如作为线性和非线性偏微分方程的数据到解决图。具体而言，我们研究了深神经操作员和广义多项式混乱（GPC）操作员的近似速率，用于无线性，可分开的希尔伯特空间之间的非线性，全态图。假定功能空间的运算符和输出通过稳定的仿射表示系统进行参数化。可接受的表示系统包括正常基础，RIESZ底座或所考虑的空间的合适的紧密框架。建立了代数表达速率界限，为具有有限的Sobolev或BESOV规律性的范围内的深层神经和GPC操作员替代物都作用于可分离的Hilbert空间和拟合图表的范围。我们通过表达速率界限来说明抽象速率界限的系数到测序图，用于圆环上线性椭圆形PDE。

实时估计实际对象深度是各种自主系统任务（例如3D重建，场景理解和状况评估）的重要模块。在机器学习的最后十年中，将深度学习方法的广泛部署到计算机视觉任务中产生了成功，从而成功地从简单的RGB模式中实现了现实的深度综合。这些模型大多数基于配对的RGB深度数据和/或视频序列和立体声图像的可用性。到目前为止，缺乏序列，立体声数据和RGB深度对使深度估计成为完全无监督的单图像转移问题，到目前为止几乎没有探索过。这项研究以生成神经网络领域的最新进展为基础，以建立完全无监督的单发深度估计。使用Wasserstein-1距离（一种新型的感知重建项和手工制作的图像过滤器）实现并同时优化了两个用于RGB至深度和深度RGB传输的发电机。我们使用工业表面深度数据以及德克萨斯州3D面部识别数据库，人类肖像的Celebamask-HQ数据库和记录人体深度的超现实数据集来全面评估模型。对于每个评估数据集，与最先进的单图像转移方法相比，提出的方法显示出深度准确性的显着提高。

On general regular simplicial partitions $\mathcal{T}$ of bounded polytopal domains $\Omega \subset \mathbb{R}^d$, $d\in\{2,3\}$, we construct \emph{exact neural network (NN) emulations} of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical ``Raviart-Thomas element'', and the ``N\'{e}d\'{e}lec edge element''. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions $\mathcal{T}$ of $\Omega$ are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension $d\geq 2$. Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra $\Omega \subset \mathbb{R}^3$. They are thus an essential ingredient in the application of e.g., the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO) methods.

我们在无限尺寸空间之间构建深度操作网络（ONET），其以指数收敛率的指数到椭圆二阶PDE的系数到溶液映射率。特别是，我们考虑在$ -dimimension周期域中设置的问题，$ d = 1,2，\ dots $，以及分析右手边和系数。我们的分析包括扩散反应问题，参数扩散方程和椭圆体系，例如异质材料的线性各向同性插座。我们利用了解决方案是分析的边值问题的谱串联方法的指数趋同。在本周期性和分析环境中，这是经典椭圆规则的。在[陈和陈，1993]和[Lu等人，2021]的oneet分支和主干构建中，我们展示了深度one的存在，它模拟了溶液映射为精确度$ \ varepsilon> 0 $在$ h ^ 1 $ norm，均匀地通过系数集。我们证明了在某些$ \ kappa> 0 $的oneet中的神经网络具有尺寸$ \ mathcal {o}（\ log | \ log（\ varepsilon）\ reval | ^ \ kappa），具体取决于物理空间维度。

对于人造深神经网络，我们证明了分析函数的表达率$ f：\ mathbb {r} ^ d \ to \ mathbb {r} $中的$ l ^ 2（\ mathbb {r} ^ d，\ gamma_d ）$ down $ d \ in {\ mathbb {n}} \ cup \ {\ idty \} $。 $ \ gamma_d $ denot $ \ mathbb {r} ^ d $的高斯产品概率测量。我们特别考虑relu和relu $ {} ^ $ y ^ $ yrucations for Integer $ k \ geq 2 $。对于$ d \ in \ mathbb {n} $，我们显示了$ l ^ 2（\ mathbb {r} ^ d，\ gamma_d）$的指数融合率。在$ d = \ infty $，在$ f：\ mathbb {r} ^ {\ mathbb {r}} \ to \ mathbb {r} $的适当平滑和稀疏假设下，用$ \ gamma_ \ idty $表示$ \ mathbb {r} ^ {\ mathbb {n}} $的无限（高斯）产品测量值，我们证明了$ l ^ 2（\ mathbb {r} ^ {\ mathbb { n}}，\ gamma_ \ idty）$。该速率仅取决于（分析延续）的量化全阵列（分析延续）地图$ f $到$ \ mathbb {c} ^ d $中的条带产品。作为应用程序，我们将深度Relu-NNS的表达率界限进行了椭圆PDE的响应曲面与Log-Gaussian随机场输入。

用于量化大型内燃机的圆柱衬里的最先进的方法，需要拆卸和切割检查的衬里。接下来是基于实验室的高分辨率显微表面深度测量，该测量基于轴承载荷曲线（也称为Abbott-Firestone曲线），对磨损进行了定量评估。这种参考方法具有破坏性，耗时且昂贵。此处介绍的研究的目的是开发无损但可靠的方法来量化表面地形。提出了一个新型的机器学习框架，该框架允许预测代表衬里表面反射RGB图像的深度轮廓的轴承载荷曲线。这些图像可以使用简单的手持显微镜收集。涉及两个神经网络模块的联合深度学习方法也优化了表面粗糙度参数的预测质量。使用定制数据库对网络堆栈进行训练，该数据库包含422个完美对齐的深度轮廓和大型气体发动机衬里的反射图像对。观察到的方法的成功表明，其在服务过程中对发动机进行现场磨损评估的巨大潜力。

Deep-learning of artificial neural networks (ANNs) is creating highly functional tools that are, unfortunately, as hard to interpret as their natural counterparts. While it is possible to identify functional modules in natural brains using technologies such as fMRI, we do not have at our disposal similarly robust methods for artificial neural networks. Ideally, understanding which parts of an artificial neural network perform what function might help us to address a number of vexing problems in ANN research, such as catastrophic forgetting and overfitting. Furthermore, revealing a network's modularity could improve our trust in them by making these black boxes more transparent. Here we introduce a new information-theoretic concept that proves useful in understanding and analyzing a network's functional modularity: the relay information $I_R$. The relay information measures how much information groups of neurons that participate in a particular function (modules) relay from inputs to outputs. Combined with a greedy search algorithm, relay information can be used to {\em identify} computational modules in neural networks. We also show that the functionality of modules correlates with the amount of relay information they carry.

Cashews are grown by over 3 million smallholders in more than 40 countries worldwide as a principal source of income. As the third largest cashew producer in Africa, Benin has nearly 200,000 smallholder cashew growers contributing 15% of the country's national export earnings. However, a lack of information on where and how cashew trees grow across the country hinders decision-making that could support increased cashew production and poverty alleviation. By leveraging 2.4-m Planet Basemaps and 0.5-m aerial imagery, newly developed deep learning algorithms, and large-scale ground truth datasets, we successfully produced the first national map of cashew in Benin and characterized the expansion of cashew plantations between 2015 and 2021. In particular, we developed a SpatioTemporal Classification with Attention (STCA) model to map the distribution of cashew plantations, which can fully capture texture information from discriminative time steps during a growing season. We further developed a Clustering Augmented Self-supervised Temporal Classification (CASTC) model to distinguish high-density versus low-density cashew plantations by automatic feature extraction and optimized clustering. Results show that the STCA model has an overall accuracy of 80% and the CASTC model achieved an overall accuracy of 77.9%. We found that the cashew area in Benin has doubled from 2015 to 2021 with 60% of new plantation development coming from cropland or fallow land, while encroachment of cashew plantations into protected areas has increased by 70%. Only half of cashew plantations were high-density in 2021, suggesting high potential for intensification. Our study illustrates the power of combining high-resolution remote sensing imagery and state-of-the-art deep learning algorithms to better understand tree crops in the heterogeneous smallholder landscape.

Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutation entropy and complexity in order to classify paintings and images of liquid crystals. Here we find that the 2 by 2 patterns of neighboring pixels come in three types. The statistics of these types, expressed by two parameters, contains the relevant information to describe and distinguish textures. The parameters are most stable and informative for isotropic structures.

It is well known that conservative mechanical systems exhibit local oscillatory behaviours due to their elastic and gravitational potentials, which completely characterise these periodic motions together with the inertial properties of the system. The classification of these periodic behaviours and their geometric characterisation are in an on-going secular debate, which recently led to the so-called eigenmanifold theory. The eigenmanifold characterises nonlinear oscillations as a generalisation of linear eigenspaces. With the motivation of performing periodic tasks efficiently, we use tools coming from this theory to construct an optimization problem aimed at inducing desired closed-loop oscillations through a state feedback law. We solve the constructed optimization problem via gradient-descent methods involving neural networks. Extensive simulations show the validity of the approach.