We introduce a novel gated recurrent unit (GRU) with a weighted time-delay feedback mechanism in order to improve the modeling of long-term dependencies in sequential data. This model is a discretized version of a continuous-time formulation of a recurrent unit, where the dynamics are governed by delay differential equations (DDEs). By considering a suitable time-discretization scheme, we propose $\tau$-GRU, a discrete-time gated recurrent unit with delay. We prove the existence and uniqueness of solutions for the continuous-time model, and we demonstrate that the proposed feedback mechanism can help improve the modeling of long-term dependencies. Our empirical results show that $\tau$-GRU can converge faster and generalize better than state-of-the-art recurrent units and gated recurrent architectures on a range of tasks, including time-series classification, human activity recognition, and speech recognition.

We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where in each iteration a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to employ indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm needs to address an infeasibility issue -- the linearized equality constraints and trust-region constraints might lead to infeasible SQP subproblems. In this regard, we propose an \textit{adaptive relaxation technique} to compute the trial step that consists of a normal step and a tangential step. To control the lengths of the two steps, we adaptively decompose the trust-region radius into two segments based on the proportions of the feasibility and optimality residuals to the full KKT residual. The normal step has a closed form, while the tangential step is solved from a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish the global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.

我们引入了一种实用方法，以实施由神经网络（NNS）定义的功能的线性偏微分方程（PDE）约束，直至所需的公差。通过将隐式函数定理的可区分物理和应用中的应用结合到NN模型中，我们开发了可区分的PDE受限的NN层。在培训期间，我们的模型学习了一个功能系列，每个功能都定义了从PDE参数到PDE解决方案的映射。在推理时，该模型通过解决PDE受限的优化问题来发现学到家族中功能的最佳线性组合。我们的方法提供了有关感兴趣领域的连续解决方案，这些解决方案完全满足了所需的物理约束。我们的结果表明，与对无约束目标的训练相比，将硬约束直接纳入NN体系结构的测试错误要低得多。

物理知识的神经网络（PINN）将问题领域的物理知识作为对损失函数的软限制，但最近的工作表明这可能导致优化困难。在这里，我们研究了搭配点的位置对这些模型训练性的影响。我们发现，随着训练的进行，可以通过适应搭配点的位置来显着提高香草·皮恩的性能。具体而言，我们提出了一种新型的自适应搭配方案，该方案逐渐将更多的搭配点（不增加数量）分配给模型正在造成更高误差的区域（基于域中损失函数的梯度）。加上在任何优化失速过程中对训练的明智重新启动（通过简单地重新采样搭配点以调整损失景观）会导致预测错误的更好估计。我们提出了一些问题的结果，包括具有不同强迫函数的2D泊松和扩散 - 辅助系统。我们发现，针对这些问题的训练香草PINN可以导致解决方案中的预测误差高达70％，尤其是在低搭配点的状态下。相比之下，我们的自适应方案可以达到较小误差的顺序，其计算复杂性与基线相似。此外，我们发现自适应方法始终如一地执行PAR或比香草Pinn方法稍好，即使对于大型搭配点方案也是如此。所有实验的代码都是开源的。

训练大型神经网络（NN）模型需要广泛的记忆资源，而激活压缩训练（ACT）是减少训练记忆足迹的一种有前途的方法。本文介绍了GACT，这是一个ACT框架，旨在支持具有有限域知识的通用NN体系结构的广泛机器学习任务。通过分析ACT近似梯度的线性化版本，我们证明了GACT的收敛性，而没有有关操作员类型或模型体系结构的先验知识。为了使训练保持稳定，我们提出了一种算法，该算法通过估计运行时对梯度的影响来决定每个张量的压缩比。我们将GACT实施为Pytorch库，很容易适用于任何NN体系结构。GACT将卷积NN，变压器和图形NNS的激活记忆降低到8.1倍，从而使4.2倍至24.7倍的训练能够较大，而精度损失可忽略不计。

我们应用随机顺序二次编程（STOSQP）算法来求解受约束的非线性优化问题，在该问题是随机的，并且约束是确定性的。我们研究了一个完全随机的设置，其中每次迭代中只有一个样本可用于估计物镜的梯度和黑森州。我们允许stosqp选择一个随机架子$ \ bar {\ alpha} _t $适应性，使得$ \ beta_t \ leq \ leq \ bar {\ alpha} _t \ leq \ leq \ beta_t+beta_t+\ chi_t+\ chi_t $，wither = o（\ beta_t）$是预定的确定性序列。我们还允许STOSQP通过随机迭代求解器（例如，使用草图和项目方法）求解牛顿系统。而且我们不需要不精确的牛顿方向的近似误差即可消失。对于这个一般的STOSQP框架，我们建立了其最后一次迭代的渐近收敛速率，最差的案例迭代复杂性是副产品。我们执行统计推断。特别是，有了适当的衰减$ \ beta_t，\ chi_t $，我们表明：（i）STOSQP方案最多可以采用$ o（1/\ epsilon^4）$ iterations $ iterations $ iTerations以实现$ \ epsilon $ -Stationarity; （ii）几乎毫无疑问，$ \ |（x_t -x^\ star，\ lambda_t- \ lambda^\ star）\ | | = o（\ sqrt {\ beta_t \ log（1/\ beta_t）}）+o（\ chi_t/\ beta_t）$，其中$（x_t，\ lambda_t）$是primal-dimal-dimal-dialal-dialal-dialal-dual stosqp itselmate; （iii）序列$ 1/\ sqrt {\ beta_t} \ cdot（x_t -x^\ star，\ lambda_t- \ lambda_t- \ lambda^\ star）$收敛到平均零高斯分布，具有非琐事的共价矩阵。此外，我们建立了$（x_t，\ lambda_t）$的Berry-Esseen，以定量地测量其分布功能的收敛性。我们还为协方差矩阵提供了实用的估计器，可以使用iTerates $ \ {（x_t，\ lambda_t）\} _ t $构建$（x^\ star，\ lambda^\ star）$的置信区间（x^\ star，\ lambda^\ star）$。我们的定理使用最可爱的测试集中的非线性问题验证。

估计大规模森林AGB和精细的空间决议对于温室气体会计，监测和验证工作以减轻气候变化的范围变得越来越重要。机载LiDAR对于在包括AGB在内的森林结构的属性建模非常有价值，但大多数LiDAR收集都发生在涵盖不规则，不连续的足迹的本地或区域尺度上，导致不同景观细分市场在各个时间点进行拼布。在这里，作为纽约州（美国）全州森林碳评估的一部分，我们解决了利用激光雷达拼布在景观尺度上的雷达拼凑而成的障碍，包括选择培训数据，对预测的区域或覆盖范围的特定模式的调查错误，并绘制与多个量表的现场清单一致。三种机器学习算法和一个集合模型经过FIA场测量，空气传播的激光雷达和地形，气候和心形地理训练。使用一组严格的地块选择标准，选择了801个FIA图，并从17个叶子覆盖范围（2014-2019）的拼布中绘制的共同定位的点云（2014-2019）。我们的合奏模型用于在预测定义的适用性区域（占激光雷达覆盖率的98％）内生成30 m AGB的预测表面，并将所得的AGB图与FIA绘图级别和面积估计值进行比较。我们的模型总体准确（％RMSE 22-45％; MAE 11.6-29.4 mg ha $^{ - 1} $; me 2.4-6.3 mg ha $^{ - 1} $），解释了73-80％的领域 - 观察到的变化，并得出与FIA基于设计的估计值一致的估计值（FIA 95％CI中的估计值的89％）。我们分享实用的解决方案，以使用LIDAR的时空拼布面临的挑战来满足不断增长的AGB映射需求，以支持森林碳会计和生态系统中的应用。

Novel plant communities reshape landscapes and pose challenges for land cover classification and mapping that can constrain research and stewardship efforts. In the US Northeast, emergence of low-statured woody vegetation, or shrublands, instead of secondary forests in post-agricultural landscapes is well-documented by field studies, but poorly understood from a landscape perspective, which limits the ability to systematically study and manage these lands. To address gaps in classification/mapping of low-statured cover types where they have been historically rare, we developed models to predict shrubland distributions at 30m resolution across New York State (NYS), using a stacked ensemble combining a random forest, gradient boosting machine, and artificial neural network to integrate remote sensing of structural (airborne LIDAR) and optical (satellite imagery) properties of vegetation cover. We first classified a 1m canopy height model (CHM), derived from a patchwork of available LIDAR coverages, to define shrubland presence/absence. Next, these non-contiguous maps were used to train a model ensemble based on temporally-segmented imagery to predict shrubland probability for the entire study landscape (NYS). Approximately 2.5% of the CHM coverage area was classified as shrubland. Models using Landsat predictors trained on the classified CHM were effective at identifying shrubland (test set AUC=0.893, real-world AUC=0.904), in discriminating between shrub/young forest and other cover classes, and produced qualitatively sensible maps, even when extending beyond the original training data. Our results suggest that incorporation of airborne LiDAR, even from a discontinuous patchwork of coverages, can improve land cover classification of historically rare but increasingly prevalent shrubland habitats across broader areas.

We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as Subsampled Newton and Newton Sketch. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We propose to address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all the past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme exhibits local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the method, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still exhibits a superlinear convergence rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.

物理建模对于许多现代科学和工程应用至关重要。从数据科学或机器学习的角度来看，更多的域 - 不可吻合，数据驱动的模型是普遍的，物理知识 - 通常表示为微分方程 - 很有价值，因为它与数据是互补的，并且可能有可能帮助克服问题例如数据稀疏性，噪音和不准确性。在这项工作中，我们提出了一个简单但功能强大且通用的框架 - 自动构建物理学，可以将各种微分方程集成到高斯流程（GPS）中，以增强预测准确性和不确定性量化。这些方程可以是线性或非线性，空间，时间或时空，与未知的源术语完全或不完整，等等。基于内核分化，我们在示例目标函数，方程相关的衍生物和潜在源函数之前构建了GP，这些函数全部来自多元高斯分布。采样值被馈送到两个可能性：一个以适合观测值，另一个符合方程式。我们使用美白方法来逃避采样函数值和内核参数之间的强依赖性，并开发出一种随机变分学习算法。在模拟和几个现实世界应用中，即使使用粗糙的，不完整的方程式，自动元素都显示出对香草GPS的改进。