Neural operators, which emerge as implicit solution operators of hidden governing equations, have recently become popular tools for learning responses of complex real-world physical systems. Nevertheless, the majority of neural operator applications has thus far been data-driven, which neglects the intrinsic preservation of fundamental physical laws in data. In this paper, we introduce a novel integral neural operator architecture, to learn physical models with fundamental conservation laws automatically guaranteed. In particular, by replacing the frame-dependent position information with its invariant counterpart in the kernel space, the proposed neural operator is by design translation- and rotation-invariant, and consequently abides by the conservation laws of linear and angular momentums. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental datasets, and show that, by automatically satisfying these essential physical laws, our learned neural operator is not only generalizable in handling translated and rotated datasets, but also achieves state-of-the-art accuracy and efficiency as compared to baseline neural operator models.
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标准的神经网络可以近似一般的非线性操作员,要么通过数学运算符的组合(例如,在对流 - 扩散反应部分微分方程中)的组合,要么仅仅是黑匣子,例如黑匣子,例如一个系统系统。第一个神经操作员是基于严格的近似理论于2019年提出的深层操作员网络(DeepOnet)。从那时起,已经发布了其他一些较少的一般操作员,例如,基于图神经网络或傅立叶变换。对于黑匣子系统,对神经操作员的培训仅是数据驱动的,但是如果知道管理方程式可以在培训期间将其纳入损失功能,以开发物理知识的神经操作员。神经操作员可以用作设计问题,不确定性量化,自主系统以及几乎任何需要实时推断的应用程序中的代替代物。此外,通过将它们与相对轻的训练耦合,可以将独立的预训练deponets用作复杂多物理系统的组成部分。在这里,我们介绍了Deponet,傅立叶神经操作员和图神经操作员的评论,以及适当的扩展功能扩展,并突出显示它们在计算机械师中的各种应用中的实用性,包括多孔媒体,流体力学和固体机制, 。
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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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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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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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部分微分方程(PDE)在许多复杂动态过程的数学建模中发挥着主导作用。解决这些PDE通常需要预定的计算成本,特别是当必须对不同的参数或条件进行多次评估时。在培训之后,神经运营商可以比传统的PDE溶剂更快地提供PDES解决方案。在这项工作中,检查两个神经运营商的不变性属性和计算复杂性,用于标量数量的运输PDE。基于图形内核网络(GKN)的神经运算符在图形结构数据上运行,以合并非识别依赖性。在这里,我们提出了改进的GKN制定以实现帧不变性。传染媒介云神经网络(VCNN)是一个具有嵌入式帧不变性的替代神经运算符,可在点云数据上运行。基于GKN的神经运营商与VCNN相比,略微更好地预测性能。然而,GKN需要过度高的计算成本,与VCNN的线性增加相比,随着越来越多的离散物对象而直角增加。
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监督运营商学习是一种新兴机器学习范例,用于建模时空动态系统的演变和近似功能数据之间的一般黑盒关系的应用。我们提出了一种新颖的操作员学习方法,LOCA(学习操作员耦合注意力),激励了最近的注意机制的成功。在我们的体系结构中,输入函数被映射到有限的一组特征,然后按照依赖于输出查询位置的注意重量平均。通过将这些注意重量与积分变换一起耦合,LOCA能够明确地学习目标输出功能中的相关性,使我们能够近似非线性运算符,即使训练集测量中的输出功能的数量非常小。我们的配方伴随着拟议模型的普遍表现力的严格近似理论保证。经验上,我们在涉及普通和部分微分方程的系统管理的若干操作员学习场景中,评估LOCA的表现,以及黑盒气候预测问题。通过这些场景,我们展示了最先进的准确性,对噪声输入数据的鲁棒性以及在测试数据集上始终如一的错误传播,即使对于分发超出预测任务。
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Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for the current operator learning methods is very high. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between function spaces. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to how we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization.
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Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems.
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部分微分方程(PDE)参见在科学和工程中的广泛使用,以将物理过程的模拟描述为标量和向量场随着时间的推移相互作用和协调。由于其标准解决方案方法的计算昂贵性质,神经PDE代理已成为加速这些模拟的积极研究主题。但是,当前的方法并未明确考虑不同字段及其内部组件之间的关系,这些关系通常是相关的。查看此类相关场的时间演变通过多活动场的镜头,使我们能够克服这些局限性。多胎场由标量,矢量以及高阶组成部分组成,例如双分数和三分分射线。 Clifford代数可以描述它们的代数特性,例如乘法,加法和其他算术操作。据我们所知,本文介绍了此类多人表示的首次使用以及Clifford的卷积和Clifford Fourier在深度学习的背景下的转换。由此产生的Clifford神经层普遍适用,并将在流体动力学,天气预报和一般物理系统的建模领域中直接使用。我们通过经验评估克利福德神经层的好处,通过在二维Navier-Stokes和天气建模任务以及三维Maxwell方程式上取代其Clifford对应物中常见的神经PDE代理中的卷积和傅立叶操作。克利福德神经层始终提高测试神经PDE代理的概括能力。
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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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群体模棱两可(例如,SE(3)均衡性)是科学的关键物理对称性,从经典和量子物理学到计算生物学。它可以在任意参考转换下实现强大而准确的预测。鉴于此,已经为将这种对称性编码为深神经网络而做出了巨大的努力,该网络已被证明可以提高下游任务的概括性能和数据效率。构建模棱两可的神经网络通常会带来高计算成本以确保表现力。因此,如何更好地折衷表现力和计算效率在模棱两可的深度学习模型的设计中起着核心作用。在本文中,我们提出了一个框架来构建可以有效地近似几何量的se(3)等效图神经网络。受差异几何形状和物理学的启发,我们向图形神经网络介绍了局部完整帧,因此可以将以给定订单的张量信息投射到框架上。构建本地框架以形成正常基础,以避免方向变性并确保完整性。由于框架仅是由跨产品操作构建的,因此我们的方法在计算上是有效的。我们在两个任务上评估我们的方法:牛顿力学建模和平衡分子构象的产生。广泛的实验结果表明,我们的模型在两种类型的数据集中达到了最佳或竞争性能。
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Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient in practical applications. However, one critical issue is that training neural operator models require large amount of ground truth data, which usually comes from the slow numerical solvers. In this paper, we propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. Training data is augmented naturally through the physical properties of differential equations such as linearity and translation. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.
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包括协调性信息,例如位置,力,速度或旋转在计算物理和化学中的许多任务中是重要的。我们介绍了概括了等级图形网络的可控e(3)的等值图形神经网络(Segnns),使得节点和边缘属性不限于不变的标量,而是可以包含相协同信息,例如矢量或张量。该模型由可操纵的MLP组成,能够在消息和更新功能中包含几何和物理信息。通过可操纵节点属性的定义,MLP提供了一种新的Activation函数,以便与可转向功能字段一般使用。我们讨论我们的镜头通过等级的非线性卷曲镜头讨论我们的相关工作,进一步允许我们引脚点点的成功组件:非线性消息聚集在经典线性(可操纵)点卷积上改善;可操纵的消息在最近发送不变性消息的最近的等价图形网络上。我们展示了我们对计算物理学和化学的若干任务的方法的有效性,并提供了广泛的消融研究。
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Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as $\textit{geometric graphs}$, $\textit{i.e.}$, graphs with nodes positioned in the Euclidean space given an $\textit{arbitrarily}$ chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as $\textit{Galilean invariance}$. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.
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深度学习替代模型已显示出在解决部分微分方程(PDE)方面的希望。其中,傅立叶神经操作员(FNO)达到了良好的准确性,并且与数值求解器(例如流体流量)上的数值求解器相比要快得多。但是,FNO使用快速傅立叶变换(FFT),该变换仅限于具有均匀网格的矩形域。在这项工作中,我们提出了一个新框架,即Geo-Fno,以解决任意几何形状的PDE。 Geo-FNO学会将可能不规则的输入(物理)结构域变形为具有均匀网格的潜在空间。具有FFT的FNO模型应用于潜在空间。所得的GEO-FNO模型既具有FFT的计算效率,也具有处理任意几何形状的灵活性。我们的Geo-FNO在其输入格式,,即点云,网格和设计参数方面也很灵活。我们考虑了各种PDE,例如弹性,可塑性,Euler和Navier-Stokes方程,以及正向建模和逆设计问题。与标准数值求解器相比,与标准数值求解器相比,Geo-fno的价格比标准数值求解器快两倍,与在现有基于ML的PDE求解器(如标准FNO)上进行直接插值相比,Geo-fno更准确。
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事实证明,与对称性的对称性在深度学习研究中是一种强大的归纳偏见。关于网格处理的最新著作集中在各种天然对称性上,包括翻译,旋转,缩放,节点排列和仪表变换。迄今为止,没有现有的体系结构与所有这些转换都不相同。在本文中,我们提出了一个基于注意力的网格数据的架构,该体系结构与上述所有转换相似。我们的管道依赖于相对切向特征的使用:一种简单,有效,等效性的替代品,可作为输入作为输入。有关浮士德和TOSCA数据集的实验证实,我们提出的架构在这些基准测试中的性能提高了,并且确实是对各种本地/全球转换的均等,因此具有强大的功能。
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作为在受边界价值约束下的部分微分方程(PDE)的经典数值求解器的替代方案,人们对研究可以有效解决此类问题的神经网络引起了人们的兴趣。在这项工作中,我们使用图神经网络(GNN)和光谱图卷积为两个不同时间独立的PDE设计了一个通用解决方案操作员。我们从有限元求解器的模拟数据上训练网络,以了解各种形状和不均匀性。与以前的作品相反,我们专注于受过训练的操作员概括以前看不见的情况的能力。具体而言,我们测试对不同形状和解决方案叠加的网格的概括,以确保不同数量的不均匀性。我们发现,在有限元网格中有很大变化的不同数据集进行培训是在所有情况下都能实现良好概括结果的关键要素。因此,我们认为GNN可以用来学习在一系列属性上概括并生成的解决方案的解决方案运算符,并比通用求解器快得多。我们可以公开可用的数据集可以使用并扩展,以验证这些模型在不同条件下的鲁棒性。
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定义网格上卷积的常用方法是将它们作为图形解释并应用图形卷积网络(GCN)。这种GCNS利用各向同性核,因此对顶点的相对取向不敏感,从而对整个网格的几何形状。我们提出了规范的等分性网状CNN,它概括了GCNS施加各向异性仪表等级核。由于产生的特征携带方向信息,我们引入了通过网格边缘并行传输特征来定义的几何消息传递方案。我们的实验验证了常规GCN和其他方法的提出模型的显着提高的表达性。
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Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical meaning, and guarantee the existence and the uniqueness of the PDE solution. Current neural-network based approaches that aim to solve PDEs rely only on training data to help the model learn BCs implicitly. There is no guarantee of BC satisfaction by these models during evaluation. In this work, we propose Boundary enforcing Operator Network (BOON) that enables the BC satisfaction of neural operators by making structural changes to the operator kernel. We provide our refinement procedure, and demonstrate the satisfaction of physics-based BCs, e.g. Dirichlet, Neumann, and periodic by the solutions obtained by BOON. Numerical experiments based on multiple PDEs with a wide variety of applications indicate that the proposed approach ensures satisfaction of BCs, and leads to more accurate solutions over the entire domain. The proposed correction method exhibits a (2X-20X) improvement over a given operator model in relative $L^2$ error (0.000084 relative $L^2$ error for Burgers' equation).
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