This work builds on the models and concepts presented in part 1 to learn approximate dictionary representations of Koopman operators from data. Part I of this paper presented a methodology for arguing the subspace invariance of a Koopman dictionary. This methodology was demonstrated on the state-inclusive logistic lifting (SILL) basis. This is an affine basis augmented with conjunctive logistic functions. The SILL dictionary's nonlinear functions are homogeneous, a norm in data-driven dictionary learning of Koopman operators. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, Extended Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators. Part 2 of this paper will extend this explanation by demonstrating the subspace invariant of heterogeneous dictionaries and presenting a head-to-head numerical comparison of deepDMD and low-parameter heterogeneous dictionary learning.

Koopman运算符将非线性动力学模型为线性动力学系统，该系统作用于非线性函数作为状态。这种非标准状态通常被称为可观察到的koopman，通常通过从\ textit {dictionary}绘制的函数的叠加来近似数值。广泛使用的算法是\ textit {扩展动态模式分解}，其中字典函数是从固定的均匀函数类中绘制的。最近，深度学习与EDMD相结合已被用来通过称为“深度动态模式分解（DEEPDMD）”的算法学习新的字典函数。学到的表示（1）都准确地模型，并且（2）与原始非线性系统的尺寸相当地缩放。在本文中，我们从deepDMD分析了学习的词典，并探索了其强劲性能的理论基础。我们发现了一类新型的字典函数，以近似Koopman可观察结果。这些字典函数的错误分析表明它们满足子空间近似的属性，我们将其定义为统一的有限近似闭合。我们发现，从不同类别的非线性函数绘制的异质词典函数的结构化混合达到了与DEEPDMD相同的精度和尺寸缩放。该混合词典以降低参数的数量级来进行，同时保持几何可解释性。我们的结果提供了一个假设，可以解释深度神经网络在学习数值近似值对Koopman操作员的成功。

Cloud computing holds the promise of reduced costs through economies of scale. To realize this promise, cloud computing vendors typically solve sequential resource allocation problems, where customer workloads are packed on shared hardware. Virtual machines (VM) form the foundation of modern cloud computing as they help logically abstract user compute from shared physical infrastructure. Traditionally, VM packing problems are solved by predicting demand, followed by a Model Predictive Control (MPC) optimization over a future horizon. We introduce an approximate formulation of an industrial VM packing problem as an MILP with soft-constraints parameterized by the predictions. Recently, predict-and-optimize (PnO) was proposed for end-to-end training of prediction models by back-propagating the cost of decisions through the optimization problem. But, PnO is unable to scale to the large prediction horizons prevalent in cloud computing. To tackle this issue, we propose the Predict-and-Critic (PnC) framework that outperforms PnO with just a two-step horizon by leveraging reinforcement learning. PnC jointly trains a prediction model and a terminal Q function that approximates cost-to-go over a long horizon, by back-propagating the cost of decisions through the optimization problem \emph{and from the future}. The terminal Q function allows us to solve a much smaller two-step horizon optimization problem than the multi-step horizon necessary in PnO. We evaluate PnO and the PnC framework on two datasets, three workloads, and with disturbances not modeled in the optimization problem. We find that PnC significantly improves decision quality over PnO, even when the optimization problem is not a perfect representation of reality. We also find that hardening the soft constraints of the MILP and back-propagating through the constraints improves decision quality for both PnO and PnC.

We consider the problem of multi-agent navigation and collision avoidance when observations are limited to the local neighborhood of each agent. We propose InforMARL, a novel architecture for multi-agent reinforcement learning (MARL) which uses local information intelligently to compute paths for all the agents in a decentralized manner. Specifically, InforMARL aggregates information about the local neighborhood of agents for both the actor and the critic using a graph neural network and can be used in conjunction with any standard MARL algorithm. We show that (1) in training, InforMARL has better sample efficiency and performance than baseline approaches, despite using less information, and (2) in testing, it scales well to environments with arbitrary numbers of agents and obstacles.

Rates of missing data often depend on record-keeping policies and thus may change across times and locations, even when the underlying features are comparatively stable. In this paper, we introduce the problem of Domain Adaptation under Missingness Shift (DAMS). Here, (labeled) source data and (unlabeled) target data would be exchangeable but for different missing data mechanisms. We show that when missing data indicators are available, DAMS can reduce to covariate shift. Focusing on the setting where missing data indicators are absent, we establish the following theoretical results for underreporting completely at random: (i) covariate shift is violated (adaptation is required); (ii) the optimal source predictor can perform worse on the target domain than a constant one; (iii) the optimal target predictor can be identified, even when the missingness rates themselves are not; and (iv) for linear models, a simple analytic adjustment yields consistent estimates of the optimal target parameters. In experiments on synthetic and semi-synthetic data, we demonstrate the promise of our methods when assumptions hold. Finally, we discuss a rich family of future extensions.

数据驱动的湍流建模正在经历数据科学算法和硬件开发后的兴趣激增。我们讨论了一种使用可区分物理范式的方法，该方法将已知的物理学与机器学习结合起来，以开发汉堡湍流的闭合模型。我们将1D汉堡系统视为一种原型测试问题，用于建模以对流为主的湍流问题中未解决的术语。我们训练一系列模型，这些模型在后验损失函数上结合了不同程度的物理假设，以测试模型在一系列系统参数（包括粘度，时间和网格分辨率）上的疗效。我们发现，以部分微分方程形式的归纳偏差的约束模型包含已知物理或现有闭合方法会产生高度数据效率，准确和可推广的模型，并且表现优于最先进的基准。以物理信息形式添加结构还为模型带来了一定程度的解释性，可能为封闭建模的未来提供了垫脚石。

机器学习（ML）是指根据大量数据预测有意义的输出或对复杂系统进行分类的计算机算法。 ML应用于各个领域，包括自然科学，工程，太空探索甚至游戏开发。本文的重点是在化学和生物海洋学领域使用机器学习。在预测全球固定氮水平，部分二氧化碳压力和其他化学特性时，ML的应用是一种有前途的工具。机器学习还用于生物海洋学领域，可从各种图像（即显微镜，流车和视频记录器），光谱仪和其他信号处理技术中检测浮游形式。此外，ML使用其声学成功地对哺乳动物进行了分类，在特定的环境中检测到濒临灭绝的哺乳动物和鱼类。最重要的是，使用环境数据，ML被证明是预测缺氧条件和有害藻华事件的有效方法，这是对环境监测的重要测量。此外，机器学习被用来为各种物种构建许多对其他研究人员有用的数据库，而创建新算法将帮助海洋研究界更好地理解海洋的化学和生物学。

本文提出了秤，这是一个一般框架，将公平原则转化为基于约束马尔可夫决策过程（CMDP）的共同表示。借助因果语言，我们的框架可以在决策过程（程序公平）以及决策（结果公平）产生的结果上构成限制。具体而言，我们表明可以将众所周知的公平原理编码为实用程序组件，非毒性组件或鳞片中心中的因果分量。我们使用涉及模拟医疗方案和现实世界中Compas数据集的一组案例研究来说明量表。实验表明，我们的框架产生了公平的政策，这些政策在单步和顺序决策方案中体现了替代公平原则。

在现实世界中的对话系统中，生成的响应必须满足几个互锁的限制：内容丰富，真实且易于控制。语言生成中的两个主要范式 - 神经语言建模和基于规则的一代 - 都难以满足这些约束。即使是最好的神经模型，也容易出现信息的幻觉和省略，而现有的基于规则的形式的形式使得很难编写既灵活又流利的语法。我们描述了对话响应产生的混合体系结构，结合了两种方法的优势。该体系结构有两个组件。首先，使用新的正式框架定义的基于规则的内容选择模型，称为数据流转导，该模型使用声明性规则将对话代理的计算（表示为数据流图）转换为代表上下文可接受响应空间的无上下文语法。其次，使用这些语法来限制神经语言模型的输出的受约束解码过程，该过程选择流利的话语。最终的系统在人类对流利，相关性和真实性的评估中的表现都优于基于规则的方法和学识渊博的方法。