Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise. For a robust recovery against noise and the choice of hyper-parameters, we introduce two new mechanisms, narrow-fit and trimming, for the coefficient support and value recovery, respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming of the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to $100\%$ noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.
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In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models which includes the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that in general the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level is above some critical threshold and the nonlinearities exhibit sufficiently fast growth. We derive explicit bounds on the critical noise threshold in the case of Gaussian white noise and provide an explicit characterization of these spurious terms in the case of trigonometric and/or polynomial model nonlinearities. However, a silver lining to this negative result is that if the data is suitably denoised (a simple moving average filter is sufficient), then we recover unconditional asymptotic consistency on the class of models with locally-Lipschitz nonlinearities. Altogether, our results reveal several important aspects of weak-form equation learning which may be used to improve future algorithms. We demonstrate our results numerically using the Lorenz system, the cubic oscillator, a viscous Burgers growth model, and a Kuramoto-Sivashinsky-type higher-order PDE.
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拟合科学数据的部分微分方程(PDE)可以用可解释的机制来代表各种以数学为导向的受试者的物理定律。从科学数据中发现PDE的数据驱动的发现蓬勃发展,作为对自然界中复杂现象进行建模的新尝试,但是当前实践的有效性通常受数据的稀缺性和现象的复杂性的限制。尤其是,从低质量数据中发现具有高度非线性系数的PDE在很大程度上已经不足。为了应对这一挑战,我们提出了一种新颖的物理学指导学习方法,该方法不仅可以编码观察知识,例如初始和边界条件,而且还包含了基本的物理原理和法律来指导模型优化。我们从经验上证明,所提出的方法对数据噪声和稀疏性更为强大,并且可以将估计误差较大。此外,我们第一次能够发现具有高度非线性系数的PDE。凭借有希望的性能,提出的方法推动了PDE的边界,这可以通过机器学习模型来进行科学发现。
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我们考虑从高噪声限制的时间序列数据中控制方程的数据驱动发现。该算法开发描述了在非线性动力学(SINDY)框架的稀疏识别的背景下避免噪声的广泛影响的方法的广泛工具包。我们提供了两个主要贡献,都集中在系统x'= f(x)中获取的嘈杂数据。首先,我们提出用于高噪声设置的广泛工具包,这是一个批判性的回归方法的扩展,从完整的库中逐步剔除剔除功能,并产生一组稀疏方程,其回归到衍生x' 。这些创新可以从高噪声时间序列数据中提取稀疏控制方程和系数(例如,增加噪声300%)。例如,它发现洛伦茨系统中的正确稀疏文库,中值系数估计误差等于1% - 3%(50%噪声),6% - 8%(100%噪声);和23% - 25%(噪音300%)。工具包中的启用模块组合成单个方法,但各个模块可以在其他方程发现方法(Sindy或不)中进行战术,以改善高噪声数据的结果。其次,我们提出了一种技术,适用于基于X'= F(X)的任何模型发现方法,以评估由于噪声数据而在非唯一解决方案的上下文中发现模型的准确性。目前,这种非唯一性可以模糊发现模型的准确性,从而造成发现方法的有效性。我们描述了一种使用线性依赖性的技术,该技术将发现的模型转换为最接近真实模型的等效形式,从而能够更准确地评估发现的模型的准确性。
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这项工作与发现物理系统的偏微分方程(PDE)有关。现有方法证明了有限观察结果的PDE识别,但未能保持令人满意的噪声性能,部分原因是由于次优估计衍生物并发现了PDE系数。我们通过引入噪音吸引物理学的机器学习(NPIML)框架来解决问题,以在任意分布后从数据中发现管理PDE。我们的建议是双重的。首先,我们提出了几个神经网络,即求解器和预选者,这些神经网络对隐藏的物理约束产生了可解释的神经表示。在经过联合训练之后,求解器网络将近似潜在的候选物,例如部分衍生物,然后将其馈送到稀疏的回归算法中,该算法最初公布了最有可能的PERSIMISIAL PDE,根据信息标准决定。其次,我们提出了基于离散的傅立叶变换(DFT)的Denoising物理信息信息网络(DPINNS),以提供一组最佳的鉴定PDE系数,以符合降低降噪变量。 Denoising Pinns的结构被划分为前沿投影网络和PINN,以前学到的求解器初始化。我们对五个规范PDE的广泛实验确认,该拟议框架为PDE发现提供了一种可靠,可解释的方法,适用于广泛的系统,可能会因噪声而复杂。
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PDE发现显示了揭示复杂物理系统的预测模型,但在测量稀疏和嘈杂时难以困难。我们介绍了一种新方法,用于PDE发现,它使用两个合理的神经网络和原始的稀疏回归算法来识别管理系统响应的隐藏动态。第一网络了解系统响应函数,而第二个网络了解一个驱动系统演进的隐藏PDE。然后,我们使用无参数稀疏回归算法从第二网络中提取隐藏PDE的人类可读形式。我们在名为PDE-读取的开源库中实现了我们的方法。我们的方法成功地识别了热,汉堡和KorteDeg-de Vries方程,具有显着的一致性。我们表明,我们的方法对稀疏性和噪音都是前所未有的强大,因此适用于现实世界的观察数据。
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封闭形式的微分方程,包括部分微分方程和高阶普通微分方程,是科学家用来建模和更好地理解自然现象的最重要工具之一。直接从数据中发现这些方程是具有挑战性的,因为它需要在数据中未观察到的各种衍生物之间建模关系(\ textit {equation-data不匹配}),并且涉及在可能的方程式的巨大空间中搜索。当前的方法对方程式的形式做出了强烈的假设,因此未能发现许多知名系统。此外,其中许多通过估计衍生物来解决方程数据不匹配,这使得它们不足以噪音且不经常采样系统。为此,我们提出了D-Cipher,这对测量工件非常健壮,可以发现新的且非常通用的微分方程类别。我们进一步设计了一种新颖的优化程序Collie,以帮助D-Cipher搜索该课程。最后,我们从经验上证明,它可以发现许多众所周知的方程,这些方程超出了当前方法的功能。
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In this paper, we introduce PDE-LEARN, a novel PDE discovery algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, $U$, to approximate the system response function and a sparse, trainable vector, $\xi$, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of $U$ and $\xi$ using a loss function that (1) makes $U$ approximate the system response function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $\xi$ characterizes, and (3) promotes sparsity in $\xi$ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.
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在科学的背景下,众所周知的格言“一张图片胜过千言万语”可能是“一个型号胜过一千个数据集”。在本手稿中,我们将Sciml软件生态系统介绍作为混合物理法律和科学模型的信息,并使用数据驱动的机器学习方法。我们描述了一个数学对象,我们表示通用微分方程(UDE),作为连接生态系统的统一框架。我们展示了各种各样的应用程序,从自动发现解决高维汉密尔顿 - Jacobi-Bellman方程的生物机制,可以通过UDE形式主义和工具进行措辞和有效地处理。我们展示了软件工具的一般性,以处理随机性,延迟和隐式约束。这使得各种SCIML应用程序变为核心训练机构的核心集,这些训练机构高度优化,稳定硬化方程,并与分布式并行性和GPU加速器兼容。
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从随机数据中揭示隐藏的动态是一个具有挑战性的问题,因为随机性参与了数据的发展。当在许多情况下没有随机数据的轨迹时,问题就变得非常复杂。在这里,我们提出了一种方法,可以根据fokker-planck(FP)方程的弱形式有效地建模随机数据的动力学,该方程控制了布朗工艺中密度函数的演变。将高斯函数作为弱形式的FP方程式的测试函数,我们将衍生物传递到高斯函数,从而将衍生物传递到高斯函数,从而通过数据的期望值近似弱形式。使用未知术语的字典表示,将线性系统构建,然后通过回归解决,从而揭示数据的未知动力学。因此,我们以弱搭配回归(WCK)方法为其三个关键组成部分命名该方法:弱形式,高斯核的搭配和回归。数值实验表明我们的方法是灵活而快速的,它在多维问题中揭示了几秒钟内的动力学,并且可以轻松地扩展到高维数据,例如20个维度。 WCR还可以正确地识别具有可变依赖性扩散和耦合漂移的复杂任务的隐藏动力学,并且性能很强,在添加噪声的情况下,在情况下达到了高精度。
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高维偏微分方程(PDE)是一种流行的数学建模工具,其应用从财务到计算化学不等。但是,用于解决这些PDE的标准数值技术通常受维度的诅咒影响。在这项工作中,我们应对这一挑战,同时着重于在具有周期性边界条件的高维域上定义的固定扩散方程。受到高维度稀疏功能近似进展的启发,我们提出了一种称为压缩傅立叶搭配的新方法。结合了压缩感应和光谱搭配的想法,我们的方法取代了结构化置式网格用蒙特卡洛采样的使用,并采用了稀疏的恢复技术,例如正交匹配的追踪和$ \ ell^1 $最小化,以近似PDE的傅立叶系数解决方案。我们进行了严格的理论分析,表明所提出的方法的近似误差与最佳$ s $ term近似(相对于傅立叶基础)与解决方案相当。我们的分析使用了最近引入的随机采样框架,我们的分析表明,在足够条件下,根据扩散系数的规律性,压缩傅立叶搭配方法相对于搭配点的数量减轻了维数的诅咒。我们还提出了数值实验,以说明稀疏和可压缩溶液近似方法的准确性和稳定性。
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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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Particle dynamics and multi-agent systems provide accurate dynamical models for studying and forecasting the behavior of complex interacting systems. They often take the form of a high-dimensional system of differential equations parameterized by an interaction kernel that models the underlying attractive or repulsive forces between agents. We consider the problem of constructing a data-based approximation of the interacting forces directly from noisy observations of the paths of the agents in time. The learned interaction kernels are then used to predict the agents behavior over a longer time interval. The approximation developed in this work uses a randomized feature algorithm and a sparse randomized feature approach. Sparsity-promoting regression provides a mechanism for pruning the randomly generated features which was observed to be beneficial when one has limited data, in particular, leading to less overfitting than other approaches. In addition, imposing sparsity reduces the kernel evaluation cost which significantly lowers the simulation cost for forecasting the multi-agent systems. Our method is applied to various examples, including first-order systems with homogeneous and heterogeneous interactions, second order homogeneous systems, and a new sheep swarming system.
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在本文中,我们考虑使用Palentir在两个和三个维度中对分段常数对象的恢复和重建,这是相对于当前最新ART的显着增强的参数级别集(PALS)模型。本文的主要贡献是一种新的PALS公式,它仅需要一个单个级别的函数来恢复具有具有多个未知对比度的分段常数对象的场景。我们的模型比当前的多对抗性,多对象问题提供了明显的优势,所有这些问题都需要多个级别集并明确估计对比度大小。给定对比度上的上限和下限,我们的方法能够以任何对比度分布恢复对象,并消除需要知道给定场景中的对比度或其值的需求。我们提供了一个迭代过程,以找到这些空间变化的对比度限制。相对于使用径向基函数(RBF)的大多数PAL方法,我们的模型利用了非异型基函数,从而扩展了给定复杂性的PAL模型可以近似的形状类别。最后,Palentir改善了作为参数识别过程一部分所需的Jacobian矩阵的条件,因此通过控制PALS扩展系数的幅度来加速优化方法,固定基本函数的中心,以及参数映射到图像映射的唯一性,由新参数化提供。我们使用X射线计算机断层扫描,弥漫性光学断层扫描(DOT),Denoising,DeonConvolution问题的2D和3D变体证明了新方法的性能。应用于实验性稀疏CT数据和具有不同类型噪声的模拟数据,以进一步验证所提出的方法。
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数据驱动的PDE的发现最近取得了巨大进展,许多规范的PDE已成功地发现了概念验证。但是,在没有事先参考的情况下,确定最合适的PDE在实际应用方面仍然具有挑战性。在这项工作中,提出了物理信息的信息标准(PIC),以合成发现的PDE的简约和精度。所提出的PIC可在不同的物理场景中七个规范的PDE上获得最新的鲁棒性,并稀疏的数据,这证实了其处理困难情况的能力。该图片还用于从实际的物理场景中从微观模拟数据中发现未开采的宏观管理方程。结果表明,发现的宏观PDE精确且简约,并满足基础的对称性,从而有助于对物理过程的理解和模拟。 PIC的命题可以在发现更广泛的物理场景中发现未透视的管理方程式中PDE发现的实际应用。
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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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光谱方法是求解部分微分方程(PDE)的科学计算的武器的重要组成部分。然而,它们的适用性和有效性在很大程度上取决于用于扩展PDE溶液的基础函数的选择。过去十年已经看到,在提供复杂职能的有效陈述方面,深入学习的出现是强烈的竞争者。在目前的工作中,我们提出了一种用谱方法结合深神经网络来解决PDE的方法。特别是,我们使用称为深度操作系统网络(DeepOnet)的深度学习技术,以识别扩展PDE解决方案的候选功能。我们已经设计了一种方法,该方法使用DeepOnet提供的候选功能作为构建具有以下属性的一组功能的起点:i)它们构成基础,2)它们是正常的,3)它们是等级的,类似于傅里叶系列或正交多项式。我们利用了我们定制的基础函数的有利属性,以研究其近似能力,并使用它们来扩展线性和非线性时间依赖性PDE的解决方案。
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从数据中发现复杂系统的基本动力是一个重要的实践主题。受限的优化算法被广泛使用并带来许多成功。但是,这种纯粹的数据驱动方法可能会在存在随机噪声的情况下会导致物理不正确,并且无法轻易通过不完整的数据来处理情况。在本文中,开发了一种具有部分观察结果的复杂湍流系统的新迭代学习算法,该算法在识别模型结构,恢复未观察到的变量和估计参数之间交替。首先,将基于因果关系的学习方法用于模型结构的稀疏识别,该方法考虑了从数据中预先学习的某些物理知识。它在应对特征之间的间接耦合方面具有独特的优势,并且与随机噪声具有鲁棒性。实用算法旨在促进高维系统的因果推断。接下来,构建了系统的非线性随机参数化,以表征未观察到的变量的时间演变。通过有效的非线性数据同化的封闭分析公式被利用以采样未观察到的变量的轨迹,然后将其视为合成观测值,以提高快速参数估计。此外,状态变量依赖性和物理约束的本地化已纳入学习过程,从而减轻维度的诅咒并防止有限的时间爆破问题。数值实验表明,新算法成功地识别模型结构并为许多具有混乱动力学,时空多尺度结构,间歇性和极端事件的复杂非线性系统提供合适的随机参数化。
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在仅给定国家的数据随着时间的推移数据时,确定系统的基本动力学的问题已经挑战了科学家数十年来的挑战。在本文中,介绍了使用机器学习对相位空间变量的{\ em更新}进行建模的方法;这是作为相空间变量的函数完成的。 (更一般而言,建模是在变量的射流空间上进行的。)该方法被证明可以准确地复制谐波振荡器,摆和Duffing振荡器的示例的动力学;在每个示例中,还可以准确恢复基础微分方程。另外,结果绝不取决于如何随时间(即定期或不规则)对数据进行采样。证明这种方法(称为“ FJET”)类似于runge-kutta(RK)数值集成方案的泰勒级数扩展产生的模型。这个类比赋予了明确揭示在建模中使用的适当功能的优势,并揭示了更新的误差估计。因此,可以将这种新方法视为通过机器学习来确定RK方案系数的一种方式。最后,在未阻尼的谐波振荡器示例中显示,更新的稳定性稳定,$ 10^9美元的$ 10^9美元的稳定性比$ 4 $ ther-ther-ther-ther-tord RK稳定。
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High-dimensional PDEs have been a longstanding computational challenge. We propose to solve highdimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs.
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