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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虽然深受深度学习在各种科学和工程问题中,由于其强大的高维非线性映射能力,但它在科学知识发现中使用有限。在这项工作中,我们提出了一种基于深度学习的框架,以发现基于高分辨率微观模拟数据的粘性重力电流的宏观控制方程,而无需先前了解基础术语。对于具有不同粘度比的两个典型方案,基于深度学习的公式完全捕获与理论上派生的术语相同的主导术语,以描述验证所提出的框架的长期渐近行为。然后获得未知的宏观方程以描述用于描述短期行为,并且最终发现了额外的深度学习补偿项。后检测的比较表明,基于深度学习的PDE实际上比理论上衍生的PDE更好地在预测长期和短期制度中预测演化粘性重力电流。此外,拟议的框架被证明是对训练的非偏见数据噪声非常稳健,这高达20%。因此,所提出的深度学习框架表明,从原始实验或模拟导致数据空间中发现了在科学语义空间中发现了未经验证的内在法律的相当潜力。
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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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物理信息的神经网络(PINN)是神经网络(NNS),它们作为神经网络本身的组成部分编码模型方程,例如部分微分方程(PDE)。如今,PINN是用于求解PDE,分数方程,积分分化方程和随机PDE的。这种新颖的方法已成为一个多任务学习框架,在该框架中,NN必须在减少PDE残差的同时拟合观察到的数据。本文对PINNS的文献进行了全面的综述:虽然该研究的主要目标是表征这些网络及其相关的优势和缺点。该综述还试图将出版物纳入更广泛的基于搭配的物理知识的神经网络,这些神经网络构成了香草·皮恩(Vanilla Pinn)以及许多其他变体,例如物理受限的神经网络(PCNN),各种HP-VPINN,变量HP-VPINN,VPINN,VPINN,变体。和保守的Pinn(CPINN)。该研究表明,大多数研究都集中在通过不同的激活功能,梯度优化技术,神经网络结构和损耗功能结构来定制PINN。尽管使用PINN的应用范围广泛,但通过证明其在某些情况下比有限元方法(FEM)等经典数值技术更可行的能力,但仍有可能的进步,最著名的是尚未解决的理论问题。
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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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Solute transport in porous media is relevant to a wide range of applications in hydrogeology, geothermal energy, underground CO2 storage, and a variety of chemical engineering systems. Due to the complexity of solute transport in heterogeneous porous media, traditional solvers require high resolution meshing and are therefore expensive computationally. This study explores the application of a mesh-free method based on deep learning to accelerate the simulation of solute transport. We employ Physics-informed Neural Networks (PiNN) to solve solute transport problems in homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that learn from large training datasets, PiNNs only leverage the strong form mathematical models to simultaneously solve for multiple dependent or independent field variables (e.g., pressure and solute concentration fields). In this study, we construct PiNN using a periodic activation function to better represent the complex physical signals (i.e., pressure) and their derivatives (i.e., velocity). Several case studies are designed with the intention of investigating the proposed PiNN's capability to handle different degrees of complexity. A manual hyperparameter tuning method is used to find the best PiNN architecture for each test case. Point-wise error and mean square error (MSE) measures are employed to assess the performance of PiNNs' predictions against the ground truth solutions obtained analytically or numerically using the finite element method. Our findings show that the predictions of PiNN are in good agreement with the ground truth solutions while reducing computational complexity and cost by, at least, three orders of magnitude.
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作为深度学习的典型{Application},物理知识的神经网络(PINN){已成功用于找到部分微分方程(PDES)的数值解决方案(PDES),但是如何提高有限准确性仍然是PINN的巨大挑战。 。在这项工作中,我们引入了一种新方法,对称性增强物理学知情的神经网络(SPINN),其中PDE的谎言对称性诱导的不变表面条件嵌入PINN的损失函数中,以提高PINN的准确性。我们分别通过两组十组独立数值实验来测试SPINN的有效性,分别用于热方程,Korteweg-De Vries(KDV)方程和潜在的汉堡{方程式},这表明Spinn的性能比PINN更好,而PINN的训练点和更简单的结构都更好神经网络。此外,我们讨论了Spinn的计算开销,以PINN的相对计算成本,并表明Spinn的训练时间没有明显的增加,甚至在某些情况下还不是PINN。
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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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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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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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The identification of material parameters occurring in constitutive models has a wide range of applications in practice. One of these applications is the monitoring and assessment of the actual condition of infrastructure buildings, as the material parameters directly reflect the resistance of the structures to external impacts. Physics-informed neural networks (PINNs) have recently emerged as a suitable method for solving inverse problems. The advantages of this method are a straightforward inclusion of observation data. Unlike grid-based methods, such as the finite element method updating (FEMU) approach, no computational grid and no interpolation of the data is required. In the current work, we aim to further develop PINNs towards the calibration of the linear-elastic constitutive model from full-field displacement and global force data in a realistic regime. We show that normalization and conditioning of the optimization problem play a crucial role in this process. Therefore, among others, we identify the material parameters for initial estimates and balance the individual terms in the loss function. In order to reduce the dependence of the identified material parameters on local errors in the displacement approximation, we base the identification not on the stress boundary conditions but instead on the global balance of internal and external work. In addition, we found that we get a better posed inverse problem if we reformulate it in terms of bulk and shear modulus instead of Young's modulus and Poisson's ratio. We demonstrate that the enhanced PINNs are capable of identifying material parameters from both experimental one-dimensional data and synthetic full-field displacement data in a realistic regime. Since displacement data measured by, e.g., a digital image correlation (DIC) system is noisy, we additionally investigate the robustness of the method to different levels of noise.
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我们提出了一种基于具有子域(CENN)的神经网络的保守能量方法,其中允许通过径向基函数(RBF),特定解决方案神经网络和通用神经网络构成满足没有边界惩罚的基本边界条件的可允许功能。与具有子域的强形式Pinn相比,接口处的损耗术语具有较低的阶数。所提出的方法的优点是效率更高,更准确,更小的近双达,而不是具有子域的强形式Pinn。所提出的方法的另一个优点是它可以基于可允许功能的特殊结构适用于复杂的几何形状。为了分析其性能,所提出的方法宫殿用于模拟代表性PDE,这些实施例包括强不连续性,奇异性,复杂边界,非线性和异质问题。此外,在处理异质问题时,它优于其他方法。
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Machine learning-based modeling of physical systems has experienced increased interest in recent years. Despite some impressive progress, there is still a lack of benchmarks for Scientific ML that are easy to use but still challenging and representative of a wide range of problems. We introduce PDEBench, a benchmark suite of time-dependent simulation tasks based on Partial Differential Equations (PDEs). PDEBench comprises both code and data to benchmark the performance of novel machine learning models against both classical numerical simulations and machine learning baselines. Our proposed set of benchmark problems contribute the following unique features: (1) A much wider range of PDEs compared to existing benchmarks, ranging from relatively common examples to more realistic and difficult problems; (2) much larger ready-to-use datasets compared to prior work, comprising multiple simulation runs across a larger number of initial and boundary conditions and PDE parameters; (3) more extensible source codes with user-friendly APIs for data generation and baseline results with popular machine learning models (FNO, U-Net, PINN, Gradient-Based Inverse Method). PDEBench allows researchers to extend the benchmark freely for their own purposes using a standardized API and to compare the performance of new models to existing baseline methods. We also propose new evaluation metrics with the aim to provide a more holistic understanding of learning methods in the context of Scientific ML. With those metrics we identify tasks which are challenging for recent ML methods and propose these tasks as future challenges for the community. The code is available at https://github.com/pdebench/PDEBench.
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We present an end-to-end framework to learn partial differential equations that brings together initial data production, selection of boundary conditions, and the use of physics-informed neural operators to solve partial differential equations that are ubiquitous in the study and modeling of physics phenomena. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our physics-informed neural operators to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled partial differential equations. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the source code, an interactive website to visualize the predictions of our physics informed neural operators, and a tutorial for their use at the Data and Learning Hub for Science.
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Physics-informed neural networks (PINNs) constitute a flexible approach to both finding solutions and identifying parameters of partial differential equations. Most works on the topic assume noiseless data, or data contaminated by weak Gaussian noise. We show that the standard PINN framework breaks down in case of non-Gaussian noise. We give a way of resolving this fundamental issue and we propose to jointly train an energy-based model (EBM) to learn the correct noise distribution. We illustrate the improved performance of our approach using multiple examples.
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当系统中有某些未知术语和隐藏的物理机制时,基于第一原理的复杂物理系统的管理方程可能会非常具有挑战性。在这项工作中,我们采用深度学习体系结构来学习基于从完全动力学模型中获取的数据的等离子体系统的流体部分微分方程(PDE)。证明了学到的多臂流体PDE可以融合诸如Landau阻尼等动力学效应。基于学习的流体闭合,数据驱动的多音阶流体建模可以很好地再现从完全动力学模型中得出的所有物理量。Landau阻尼的计算阻尼率与完全动力学的模拟和线性理论一致。用于复杂物理系统的PDE的数据驱动的流体建模可以应用于改善流体闭合并降低全球系统多规模建模的计算成本。
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Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.
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We propose characteristic-informed neural networks (CINN), a simple and efficient machine learning approach for solving forward and inverse problems involving hyperbolic PDEs. Like physics-informed neural networks (PINN), CINN is a meshless machine learning solver with universal approximation capabilities. Unlike PINN, which enforces a PDE softly via a multi-part loss function, CINN encodes the characteristics of the PDE in a general-purpose deep neural network trained with the usual MSE data-fitting regression loss and standard deep learning optimization methods. This leads to faster training and can avoid well-known pathologies of gradient descent optimization of multi-part PINN loss functions. If the characteristic ODEs can be solved exactly, which is true in important cases, the output of a CINN is an exact solution of the PDE, even at initialization, preventing the occurrence of non-physical outputs. Otherwise, the ODEs must be solved approximately, but the CINN is still trained only using a data-fitting loss function. The performance of CINN is assessed empirically in forward and inverse linear hyperbolic problems. These preliminary results indicate that CINN is able to improve on the accuracy of the baseline PINN, while being nearly twice as fast to train and avoiding non-physical solutions. Future extensions to hyperbolic PDE systems and nonlinear PDEs are also briefly discussed.
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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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Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.
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