重型模型引起了神经网络现代发展的关注。深度平衡模型(DEQ)代表具有重量趋势的无限深度神经网络,最近的研究表明了这种方法的潜力。需要迭代解决训练中的根发现问题,并建立在模型确定的基础动力学基础上,需要DEQ。在本文中,我们介绍了稳定的不变模型(SIM),这是一种新的深层模型,原理在稳定性下近似DEQ,并将动力学扩展到更一般的动力学,从而收敛到不变的集合(不受固定点的限制)。得出SIMS的关键要素是用Koopman和Perron--Frobenius操作员的光谱表示动力学的代表。该视角大致揭示了用DEQS揭示稳定的动力学,然后衍生了两个SIMS的变体。我们还提出了可以以与前馈模型相同的方式学习的SIMS的实现。我们通过实验说明了SIMS的经验表现,并证明SIMS在几个学习任务中对DEQ实现了比较或出色的表现。
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We present a new approach to modeling sequential data: the deep equilibrium model (DEQ). Motivated by an observation that the hidden layers of many existing deep sequence models converge towards some fixed point, we propose the DEQ approach that directly finds these equilibrium points via root-finding. Such a method is equivalent to running an infinite depth (weight-tied) feedforward network, but has the notable advantage that we can analytically backpropagate through the equilibrium point using implicit differentiation. Using this approach, training and prediction in these networks require only constant memory, regardless of the effective "depth" of the network. We demonstrate how DEQs can be applied to two state-of-the-art deep sequence models: self-attention transformers and trellis networks. On large-scale language modeling tasks, such as the WikiText-103 benchmark, we show that DEQs 1) often improve performance over these stateof-the-art models (for similar parameter counts); 2) have similar computational requirements to existing models; and 3) vastly reduce memory consumption (often the bottleneck for training large sequence models), demonstrating an up-to 88% memory reduction in our experiments. The code is available at https://github. com/locuslab/deq.
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神经运营商最近成为设计神经网络形式的功能空间之间的解决方案映射的流行工具。不同地,从经典的科学机器学习方法,以固定分辨率为输入参数的单个实例学习参数,神经运算符近似PDE系列的解决方案图。尽管他们取得了成功,但是神经运营商的用途迄今为止仅限于相对浅的神经网络,并限制了学习隐藏的管理法律。在这项工作中,我们提出了一种新颖的非局部神经运营商,我们将其称为非本体内核网络(NKN),即独立的分辨率,其特征在于深度神经网络,并且能够处理各种任务,例如学习管理方程和分类图片。我们的NKN源于神经网络的解释,作为离散的非局部扩散反应方程,在无限层的极限中,相当于抛物线非局部方程,其稳定性通过非本种载体微积分分析。与整体形式的神经运算符相似允许NKN捕获特征空间中的远程依赖性,而节点到节点交互的持续处理使NKNS分辨率独立于NKNS分辨率。与神经杂物中的相似性,在非本体意义上重新解释,并且层之间的稳定网络动态允许NKN的最佳参数从浅到深网络中的概括。这一事实使得能够使用浅层初始化技术。我们的测试表明,NKNS在学习管理方程和图像分类任务中占据基线方法,并概括到不同的分辨率和深度。
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Koopman运算符是无限维的运算符,可全球线性化非线性动态系统,使其光谱信息可用于理解动态。然而,Koopman运算符可以具有连续的光谱和无限维度的子空间,使得它们的光谱信息提供相当大的挑战。本文介绍了具有严格融合的数据驱动算法,用于从轨迹数据计算Koopman运算符的频谱信息。我们引入了残余动态模式分解(ResDMD),它提供了第一种用于计算普通Koopman运算符的Spectra和PseudtoStra的第一种方案,无需光谱污染。使用解析器操作员和RESDMD,我们还计算与测量保存动态系统相关的光谱度量的平滑近似。我们证明了我们的算法的显式收敛定理,即使计算连续频谱和离散频谱的密度,也可以实现高阶收敛即使是混沌系统。我们展示了在帐篷地图,高斯迭代地图,非线性摆,双摆,洛伦茨系统和11美元延长洛伦兹系统的算法。最后,我们为具有高维状态空间的动态系统提供了我们的算法的核化变体。这使我们能够计算与具有20,046维状态空间的蛋白质分子的动态相关的光谱度量,并计算出湍流流过空气的误差界限的非线性Koopman模式,其具有雷诺数为$> 10 ^ 5 $。一个295,122维的状态空间。
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学习的优化器是可以训练解决优化问题的算法。与使用从理论原则派生的简单更新规则的基线优化器(例如势头或亚当)相比,学习的优化器使用灵活,高维,非线性参数化。虽然这可能导致某些设置中的更好性能,但他们的内部工作仍然是一个谜。学习优化器如何优于一个良好的调整基线?它是否学习了现有优化技术的复杂组合,或者是实现全新的行为吗?在这项工作中,我们通过仔细分析和可视化的学习优化器来解决这些问题。我们研究了从三个不同的任务中从头开始培训的优化器,并发现他们已经了解了可解释的机制,包括:势头,渐变剪辑,学习率计划以及新形式的学习率适应形式。此外,我们展示了学习优化器的动态如何实现这些行为。我们的结果帮助阐明了对学习优化器的工作原理的先前密切了解,并建立了解释未来学习优化器的工具。
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Regularising the parameter matrices of neural networks is ubiquitous in training deep models. Typical regularisation approaches suggest initialising weights using small random values, and to penalise weights to promote sparsity. However, these widely used techniques may be less effective in certain scenarios. Here, we study the Koopman autoencoder model which includes an encoder, a Koopman operator layer, and a decoder. These models have been designed and dedicated to tackle physics-related problems with interpretable dynamics and an ability to incorporate physics-related constraints. However, the majority of existing work employs standard regularisation practices. In our work, we take a step toward augmenting Koopman autoencoders with initialisation and penalty schemes tailored for physics-related settings. Specifically, we propose the "eigeninit" initialisation scheme that samples initial Koopman operators from specific eigenvalue distributions. In addition, we suggest the "eigenloss" penalty scheme that penalises the eigenvalues of the Koopman operator during training. We demonstrate the utility of these schemes on two synthetic data sets: a driven pendulum and flow past a cylinder; and two real-world problems: ocean surface temperatures and cyclone wind fields. We find on these datasets that eigenloss and eigeninit improves the convergence rate by up to a factor of 5, and that they reduce the cumulative long-term prediction error by up to a factor of 3. Such a finding points to the utility of incorporating similar schemes as an inductive bias in other physics-related deep learning approaches.
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神经网络的经典发展主要集中在有限维欧基德空间或有限组之间的学习映射。我们提出了神经网络的概括,以学习映射无限尺寸函数空间之间的运算符。我们通过一类线性积分运算符和非线性激活函数的组成制定运营商的近似,使得组合的操作员可以近似复杂的非线性运算符。我们证明了我们建筑的普遍近似定理。此外,我们介绍了四类运算符参数化:基于图形的运算符,低秩运算符,基于多极图形的运算符和傅里叶运算符,并描述了每个用于用每个计算的高效算法。所提出的神经运营商是决议不变的:它们在底层函数空间的不同离散化之间共享相同的网络参数,并且可以用于零击超分辨率。在数值上,与现有的基于机器学习的方法,达西流程和Navier-Stokes方程相比,所提出的模型显示出卓越的性能,而与传统的PDE求解器相比,与现有的基于机器学习的方法有关的基于机器学习的方法。
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这是一门专门针对STEM学生开发的介绍性机器学习课程。我们的目标是为有兴趣的读者提供基础知识,以在自己的项目中使用机器学习,并将自己熟悉术语作为进一步阅读相关文献的基础。在这些讲义中,我们讨论受监督,无监督和强化学习。注释从没有神经网络的机器学习方法的说明开始,例如原理分析,T-SNE,聚类以及线性回归和线性分类器。我们继续介绍基本和先进的神经网络结构,例如密集的进料和常规神经网络,经常性的神经网络,受限的玻尔兹曼机器,(变性)自动编码器,生成的对抗性网络。讨论了潜在空间表示的解释性问题,并使用梦和对抗性攻击的例子。最后一部分致力于加强学习,我们在其中介绍了价值功能和政策学习的基本概念。
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时间序列数据的生成和分析与许多从经济学到流体力学的定量字段相关。在物理科学中,诸如亚稳态和连贯的组的结构,慢松弛过程,集体变量显性过渡途径或歧管流动流动的概率流动可能非常重视理解和表征系统的动力动力学和机械性质。 Deeptime是一种通用Python库,提供各种工具来估计基于时间序列数据的动态模型,包括传统的线性学习方法,例如马尔可夫状态模型(MSM),隐藏的马尔可夫模型和Koopman模型,以及内核和深度学习方法如vampnets和深msms。该库主要兼容Scikit-Searn,为这些不同的模型提供一系列估计器类,但与Scikit-Ge劳说相比,还提供了深度模型类,例如,在MSM的情况下,提供了多种分析方法来计算有趣的热力学,动力学和动态量,例如自由能,松弛时间和过渡路径。图书馆专为易于使用而设计,而且易于维护和可扩展的代码。在本文中,我们介绍了Deeptime软件的主要特征和结构。
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有效地对远程依赖性建模是序列建模的重要目标。最近,使用结构化状态空间序列(S4)层的模型在许多远程任务上实现了最先进的性能。 S4层将线性状态空间模型(SSM)与深度学习技术结合在一起,并利用HIPPO框架进行在线功能近似以实现高性能。但是,该框架导致了架构约束和计算困难,使S4方法变得复杂,可以理解和实施。我们重新审视这样的想法,即遵循河马框架对于高性能是必要的。具体而言,我们替换了许多独立的单输入单输出(SISO)SSM的库S4层与一个多输入的多输出(MIMO)SSM一起使用,并具有降低的潜在尺寸。 MIMO系统的缩小潜在维度允许使用有效的并行扫描,从而简化了将S5层应用于序列到序列转换所需的计算。此外,我们将S5 SSM的状态矩阵初始化,其近似与S4 SSMS使用的河马级矩阵近似,并表明这是MIMO设置的有效初始化。 S5与S4在远程任务上的表现相匹配,包括在远程竞技场基准的套件中平均达到82.46%,而S4的80.48%和最佳的变压器变体的61.41%。
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我们提出了一种从数据模拟动态系统的数值方法。我们使用最近引入的方法可扩展的概率近似(SPA)从欧几里德空间到凸多台的项目点,并表示在新的低维坐标中的系统的预计状态,表示其在多晶硅中的位置。然后,我们介绍特定的非线性变换,以构建多特渗透中动力学的模型,并转换回原始状态空间。为了克服投影到低维层的潜在信息损失,我们在局部延迟嵌入定理的意义上使用记忆。通过施工,我们的方法产生稳定的模型。我们说明了在各种示例上具有多个连接组件的甚至复制混沌动力学和吸引子的方法的能力。
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These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics. The main objective of these notes is to introduce a student who is familiar with concepts in linear algebra and partial differential equations to select topics in deep learning. These lecture notes exploit the strong connections between deep learning algorithms and the more conventional techniques of computational physics to achieve two goals. First, they use concepts from computational physics to develop an understanding of deep learning algorithms. Not surprisingly, many concepts in deep learning can be connected to similar concepts in computational physics, and one can utilize this connection to better understand these algorithms. Second, several novel deep learning algorithms can be used to solve challenging problems in computational physics. Thus, they offer someone who is interested in modeling a physical phenomena with a complementary set of tools.
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Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.
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基于近似基础的Koopman操作员或发电机的数据驱动的非线性动力系统模型已被证明是预测,功能学习,状态估计和控制的成功工具。众所周知,用于控制膜系统的Koopman发电机还对输入具有仿射依赖性,从而导致动力学的方便有限维双线性近似。然而,仍然存在两个主要障碍,限制了当前方法的范围,以逼近系统的koopman发电机。首先,现有方法的性能在很大程度上取决于要近似Koopman Generator的基础函数的选择;目前,目前尚无通用方法来为无法衡量保存的系统选择它们。其次,如果我们不观察到完整的状态,我们可能无法访问足够丰富的此类功能来描述动态。这是因为在有驱动时,通常使用时间延迟的可观察物的方法失败。为了解决这些问题,我们将Koopman Generator控制的可观察到的动力学写为双线性隐藏Markov模型,并使用预期最大化(EM)算法确定模型参数。 E-Step涉及标准的Kalman滤波器和更光滑,而M-Step类似于发电机的控制效果模式分解。我们在三个示例上证明了该方法的性能,包括恢复有限的Koopman-Invariant子空间,用于具有缓慢歧管的驱动系统;估计非强制性行驶方程的Koopman本征函数;仅基于提升和阻力的嘈杂观察,对流体弹球系统的模型预测控制。
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Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged as a leading data-driven embedding, as eigenfunctions of this operator provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven to be mathematically and computationally challenging. This work leverages the power of deep learning to discover representations of Koopman eigenfunctions from trajectory data of dynamical systems. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold parameterized by these eigenfunctions. In particular, we identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems that exhibit continuous spectra, ranging from the simple pendulum to nonlinear optics and broadband turbulence. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to half a century of asymptotics. In this way, we benefit from the power and generality of deep learning, while retaining the physical interpretability of Koopman embeddings.
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平衡系统是表达神经计算的有力方法。作为特殊情况,它们包括对神经科学和机器学习的最新兴趣模型,例如平衡复发性神经网络,深度平衡模型或元学习。在这里,我们提出了一个新的原则,用于学习具有时间和空间本地规则的此类系统。我们的原理将学习作为一个最不控制的问题,我们首先引入一个最佳控制器,以将系统带入解决方案状态,然后将学习定义为减少达到这种状态所需的控制量。我们表明,将学习信号纳入动力学作为最佳控制可以以先前未知的方式传输信用分配信息,避免将中间状态存储在内存中,并且不依赖无穷小的学习信号。在实践中,我们的原理可以使基于梯度的学习方法的强大绩效匹配,该方法应用于涉及复发性神经网络和元学习的一系列问题。我们的结果阐明了大脑如何学习并提供解决广泛的机器学习问题的新方法。
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这项调查的目的是介绍对深神经网络的近似特性的解释性回顾。具体而言,我们旨在了解深神经网络如何以及为什么要优于其他经典线性和非线性近似方法。这项调查包括三章。在第1章中,我们回顾了深层网络及其组成非线性结构的关键思想和概念。我们通过在解决回归和分类问题时将其作为优化问题来形式化神经网络问题。我们简要讨论用于解决优化问题的随机梯度下降算法以及用于解决优化问题的后传播公式,并解决了与神经网络性能相关的一些问题,包括选择激活功能,成本功能,过度适应问题和正则化。在第2章中,我们将重点转移到神经网络的近似理论上。我们首先介绍多项式近似中的密度概念,尤其是研究实现连续函数的Stone-WeierStrass定理。然后,在线性近似的框架内,我们回顾了馈电网络的密度和收敛速率的一些经典结果,然后在近似Sobolev函数中进行有关深网络复杂性的最新发展。在第3章中,利用非线性近似理论,我们进一步详细介绍了深度和近似网络与其他经典非线性近似方法相比的近似优势。
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Koopman运算符全球线性化非线性动力学系统及其光谱信息是分析和分解非线性动力学系统的强大工具。但是,Koopman运营商是无限维度的,计算其光谱信息是一个巨大的挑战。我们介绍了Measure-tearving扩展动态模式分解($ \ texttt {mpedmd} $),这是第一种截断方法,其特征性组件收敛到koopman运算符的光谱,以用于一般测量的动态系统。 $ \ texttt {mpedmd} $是基于正交式procrustes问题的数据驱动算法,该问题使用可观察的一般字典来强制测量Koopman运算符的截断。它具有灵活性且易于使用的任何预先存在的DMD类型方法,并且具有不同类型的数据。我们证明了$ \ texttt {mpedmd} $的融合,用于投影值和标量值光谱测量,光谱和koopman模式分解。对于延迟嵌入(Krylov子空间)的情况,我们的结果包括随着字典的大小增加,光谱测量近似值的第一个收敛速率。我们在一系列具有挑战性的示例中演示了$ \ texttt {mpedmd} $,与其他DMD型方法相比,其对噪声的稳健性提高,以及其捕获湍流边界层实验测量的能源保存和级联反应的能力,并以Reynolds的方式流动。数字$> 6 \ times 10^4 $和状态空间尺寸$> 10^5 $。
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Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by having decomposable operators, we show that the classical ``operator splitting'' numerical scheme of solving these equations can be exploited to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics contains learnable parameters, and is thus more flexible than the standard operator splitting scheme. Once trained, it enables the fast solution of the same type of PDEs. To validate the special structure inside DOSnet, we take the linear PDEs as the benchmark and give the mathematical explanation for the weight behavior. Furthermore, to demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schr\"odinger equations (NLSE) which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.
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Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.
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