本文节选自《科学》（SCIENCE ADVANCES ? 16 Feb 2022 ? Vol 8, Issue 7）Analyses of internal structures and defects in materials using physics-informed neural networks——利用物理信息神经网络分析材料内部结构和缺陷。

Analyses of internal structures and defects in materials using physics-informed neural networks

利用物理信息神经网络分析材料内部结构和缺陷

Characterizing internal structures and defects in materials is a challenging task, often requiring solutions to inverse problems with unknown topology, geometry, material properties, and nonlinear deformation.

描述材料内部结构和缺陷是一项具有挑战性的任务，通常需要解决未知拓扑、几何、材料属性和非线性变形的反问题。

Here, we present

a general framework based on physics-informed neural networks for identifying unknown geometric and material parameters.

在这里，我们提出了一个基于物理信息神经网络的通用框架，用于识别未知的几何和材料参数。

By using a mesh-free method, we parameterize the geometry of the material using a differentiable and trainable method that can identify multiple structural features.

通过使用一种无网格方法，我们使用一种可微分和可训练的方法来参数化材料的几何结构，该方法可以识别多个结构特征。

We validate this approach for materials with internal voids/inclusions using constitutive models that encompass the spectrum of linear elasticity, hyperelasticity, and plasticity.

我们使用包含线弹性、超弹性和塑性谱的本构模型，验证了这种方法对于含有内部空洞/夹杂物的材料。

We predict the size, shape, and location of the internal void/inclusion as well as the elastic modulus of the inclusion.

我们预测了内部空洞/夹杂物的尺寸、形状和位置，以及夹杂物的弹性模量。

Our general framework can be applied to other inverse problems

in different applications that involve unknown material properties and highly deformable geometries, targeting material characterization, quality assurance, and structural design.

我们的通用框架可以应用于涉及未知材料特性和高度可变形几何的不同应用中的其他反问题，针对材料特性、质量保证和结构设计。

重点词汇

characterizing具有…的特征；（characterize的现在分词形式）；塑造人物

challenging task具有挑战性的任务

parameterize确定…的参数；用参数表示

differentiable可区别的

trainable可教育的；可训练的

voids空洞；孔洞；空隙率；结点间；（void的复数）；使无效；排泄；清理场地；（void的第三人称单数）

inclusions内含物；（inclusion的复数）

constitutive有建立权的；有设立权的；有制定权的；有任命权的；组成的；构成的；成分的；基本的；本质的

encompass环绕；围绕；包围；围住；促成；实现；包括；包含

plasticity可塑性；适应力

Deep learning approaches play an increasingly substantial role in a wide range of technologies that benefit computer vision, natural language processing, and other data-rich areas of societal interest.

深度学习方法在广泛的技术领域发挥着越来越重要的作用，这些技术有利于计算机视觉、自然语言处理和其他社会感兴趣的数据丰富的领域。

Despite the evolving sophistication of data analytics and neural networks (NNs), much of this work to date has not been predicated on a large volume of scientific data, through which predictive models can be constructed using experimentally validated mechanistic inferences and laws of physics.

尽管数据分析和神经网络(NNs)的复杂性不断发展，但迄今为止，这些工作中的大部分都不是基于大量的科学数据，通过这些数据，可以使用实验验证的机械推理和物理定律来构建预测模型。

In most scientific applications, by contrast, physical conservation laws (such as those for momentum and energy) are framed by highly general, mathematical formulations [e.g., those invoking partial differential equations (PDEs) in areas such as solid mechanics, fluid mechanics, and material diffusion], along with experimental authentication by recourse to laboratory tests.

相比之下，在大多数科学应用中，物理守恒定律(如动量和能量守恒定律)是由高度通用的数学公式(例如，在固体力学、流体力学和材料扩散等领域引用偏微分方程(PDEs))以及借助实验室测试的实验验证来构建的。

重点词汇

computer vision电脑视觉；计算机视觉；计算机视觉研究

natural language自然语言

sophistication诡辩；复杂；老练；有教养

predicated宣布为；（predicate的过去式）；断定；断言；谓语的

large volume大体积

mechanistic机械论的；机械性的

inferences推理；结论；推断；（inference的复数）

conservation laws守恒律；守恒定律

invoking祈求；引起；（invoke的现在分词形式）

fluid mechanics流体力学

Emerging research reveals the profound untapped potential of physics-based, multidisciplinary, deep learning approaches with unprecedented opportunities for scientific and engineering advances in molecular analysis, design of materials with improved properties and performance in structural and functional applications, and unique pathways for the characterization of properties of materials.

新兴的研究揭示了基于物理学的多学科深度学习方法的巨大潜力，为分子分析的科学和工程进步、在结构和功能应用中改进性能和性能的材料设计以及材料性能表征的独特途径带来了前所未有的机遇。

To further realize this potential, broadly applicable methodologies in the area of NNs are needed to address a variety of issues that underpin deep learning analyses, governed by physical laws and guided by mathematical formulations.

为了进一步实现这一潜力，需要在神经网络（NNs）领域采用广泛适用的方法来解决各种问题，这些问题是深入学习分析的基础，受物理定律的制约，并以数学公式为指导。

To this end, a physics-informed deep learning approach has recently been proposed for the simulation of systems governed by physical laws that are represented by PDEs.

为此，最近提出了一种基于物理的深度学习方法，用于模拟由偏微分方程表示的物理定律所控制的系统。

重点词汇

reveals揭露；泄露；透露；显示；（reveal的第三人称单数）；显露；启示；门侧；窗侧；（汽车的）窗框；（reveal的复数）

untapped未被开发的；未被利用的

multidisciplinary多种学科结合的；多种学科的；多专业的

unprecedented无先例的；前所未闻的

broadly总地；大体上；概括地；宽阔地；敞开地；广泛地；公开地

methodologies（从事某一研究的）方法或原则；方法学；方法论；方法问题；(methodology的复数)

underpin加固…地下基础；支持；巩固；构成…的基础

governed管理；治理；支配；统治；控制；操纵；（govern的过去式和过去分词）

formulations构想；配方；公式化；规划；(formulation的复数)

to this end有鉴于此；因为这个；为此

While traditional methods based on deep learning implicitly encode these formulations by feeding training data governed by equations, this approach explicitly encodes known physical or scaling laws in the form of mathematical equations into the standard structure of NNs, formulating the so-called physics-informed NNs (PINNs).

传统的基于深度学习的方法通过输入由方程控制的训练数据来隐式编码这些公式，而该方法明确地将已知的物理或标度定律以数学方程的形式编码到神经网络的标准结构中，形成所谓的物理通知神经网络(PINNs)。

Such an approach integrates any existing knowledge expressible in terms of PDEs during the learning process, thereby markedly improving predictability while reducing the amount of data required to achieve a desired level of accuracy.

这种方法在学习过程中综合了任何可用偏微分方程表示的现有知识，从而大大提高了可预测性，同时减少了达到所需准确度水平所需的数据量。

Studies have shown the applicability of PINNs in addressing a wide spectrum of forward and inverse problems spanning disciplines such as fluid mechanics, quantum mechanics, and solid mechanics.

研究表明，PINNs在解决跨越流体力学、量子力学和固体力学等学科的广泛正问题和反问题方面具有适用性。

重点词汇

implicitly含蓄地；无疑问地；无保留地；暗示地

encode把…译成密码；把（信息；指令）编码；对生成（某种物质或行为）负责；给…编码

formulations构想；配方；公式化；规划；(formulation的复数)

encodes编码；（将文字材料）译成密码；编制成计算机语言；（encode的第三人称单数）

integrates（使）结合；成为一体；使合并；使整合；（integrate的第三人称单数）

expressible可榨出的；可表现的

predictability可预测性；可预见性；可预言

spanning以掌测量；横跨；跨越；持续；贯穿；包括；涵盖；(span的现在分词)；（指树形子图）生成的

fluid mechanics流体力学

quantum mechanics量子力学

These applications have shown promise for enhancing predictability when the amount of data is limited or when the problem is ill posed, situations in which existing methods are not likely to yield accurate and reliable results.

当数据量有限或问题不适当时，这些应用显示出增强可预测性的前景，在这种情况下，现有方法不可能产生准确可靠的结果。

This approach has been further extended to offer unique pathways to address relevant mathematical formulations, such as stochastic PDEs and fractional PDEs.

这种方法已被进一步扩展，以提供独特的途径来解决相关的数学公式，如随机偏微分方程和分数偏微分方程。

重点词汇

promise允诺；诺言；承诺；保证；可能；前途；指望；出息；答应；作出保证；有…可能；给…以指望；期待；把…许配给；宣布…

predictability可预测性；可预见性；可预言

posed造成；装模作样；假装；提出；形成；构成；摆出姿态或姿势；（pose的过去式和过去分词）

not likely不大可能；可能不；可能不会

accurate准确的；精确的；能精确击中目标的；真实的；逼真的；精密的；准确无误的；射中（或投中）目标的

reliable可靠的；可信任的；可靠的人

mathematical数学的；数学般精确的

formulations构想；配方；公式化；规划；(formulation的复数)

stochastic有可能的；或然的；随机的

fractional分数的；少量的；一小部分的；分馏的；分级的

Here, we address geometry identification problems in the field of continuum solid mechanics.

在这里，我们讨论了连续介质固体力学领域中的几何识别问题。

Geometry identification problems are a class of inverse problems of scientific, technological, and societal interest in fields as diverse as the following: safety and failure analysis of civil, mechanical, nuclear, and aeronautical structures; land, sea, and air transportations; reliability analysis in microelectronic devices; nondestructive testing of materials; and processing of engineered materials.

几何识别问题是一类涉及科学、技术和社会利益的反问题，涉及以下领域：民用、机械、核和航空结构的安全和失效分析；陆、海、空运输；微电子器件可靠性分析；材料的无损检测；以及工程材料的加工。

In a geometry identification problem, the unknown geometric features and parameters are determined in a solid material/structure given measured material response under static or dynamic loading, thereby characterizing unknown structures including internal defects or boundaries such as voids, vacancies or holes, inclusions and reinforcements, and/or cracks.

在几何识别问题中，给定静态或动态载荷下测量的材料响应，在固体材料/结构中确定未知的几何特征和参数，从而表征未知的结构，包括内部缺陷或边界，例如空隙、空位或洞、夹杂物和增强物、和/或裂纹。

Traditionally, computational algorithms for geometry identification are established on the basis of the finite element method (FEM) as the forward solver.

传统上，几何识别的计算算法是基于有限元法（FEM）作为正向求解器建立的。

Beyond the forward solver, considerable effort is required for the design and implementation of iterative algorithms for updating the estimated values of geometric parameters (see section S1 for a brief review of the algorithms), through which the discrepancy (loss) between the observed data and the results of the forward solver is minimized.

除了正向解算器，还需要花费大量精力来设计和实施迭代算法，以更新几何参数的估计值（参见第S1节对算法的简要回顾），从而将观测数据和正向解算器结果之间的差异（损失）降至最低。

However, the embedded forward FEM solver as a mesh-based method inherently brings about complications in these algorithms. The estimated geometry is updated by repeatedly remeshing the domain through iterations.

然而，嵌入式前向有限元求解器作为一种基于网格的方法固有地带来了这些算法的复杂性。通过迭代对区域进行反复的网格重构，从而更新估计的几何形状。

Alternatively, the unknown domain is embedded in a larger fixed domain while introducing an auxiliary field to track the presence of material.

或者，在引入辅助场以跟踪材料存在的同时，将未知域嵌入更大的固定域中。

The problem becomes even more challenging when large deformations (i.e., geometric nonlinearity) and nonlinear mechanical properties (i.e., highly nonlinear constitutive behavior of the solid material) are involved.

当涉及到大变形(即几何非线性)和非线性力学特性(即固体材料的高度非线性本构行为)时，这个问题变得更加具有挑战性。

These issues are still not well resolved, and available methods are cumbersome and resource intensive for deriving automated solutions to such inverse problems involving unknown geometry.

这些问题仍然没有得到很好的解决，现有的方法对于推导此类涉及未知几何体的反问题的自动解决方案来说既繁琐又占用大量资源。

重点词汇

continuum差异序列；族群相；连续统；闭联集

societal社会的；社会关系的

failure analysis故障分析；失误分析；失败分析

aeronautical航空的；航空（学）的；航空学的

transportations运输；运输工具；交通工具；流放；（transportation的复数）

microelectronic微电子的

nondestructive非破坏性的；无损的

engineered设计；制造；精心安排；策划；改变（生物体）的遗传性状；（engineer的过去式和过去分词）

characterizing具有…的特征；（characterize的现在分词形式）；塑造人物

voids空洞；孔洞；空隙率；结点间；（void的复数）；使无效；排泄；清理场地；（void的第三人称单数）

Here, we present a unique, systematic approach based on PINNs for solving geometry identification problems in continuum solid mechanics.

在这里，我们提出了一种独特的、基于PINNs的系统方法，用于解决连续介质固体力学中的几何识别问题。

This method integrates known PDEs of importance in solid mechanics with NNs, composing a unified computational framework involving both the forward solver and the inverse algorithm.

该方法将固体力学中重要的已知偏微分方程与神经网络相结合，构成一个统一的计算框架，包括正反演算法。

Notably, we propose a method for directly parameterizing the geometry of the solid in a differentiable and trainable manner.

值得注意的是，我们提出了一种以可微分和可训练的方式直接参数化实体几何形状的方法。

By using the workflow of NNs, our method can automatically update the geometry estimation through the deep learning process.

该方法利用神经网络的工作流程，通过深度学习过程自动更新几何估计。

重点词汇

systematic approach系统观点；系统研究法；系统性方针

continuum差异序列；族群相；连续统；闭联集

integrates（使）结合；成为一体；使合并；使整合；（integrate的第三人称单数）

of importance重要性；重要；具有重要性

composing构成；创作；组成；作曲；（compose的现在分词形式）；起镇静作用的

computational使用计算机的；计算机的；计算的

solver解决者；[计]解算机；[数]求解程序

differentiable可区别的

trainable可教育的；可训练的

workflow工作流程

To demonstrate the efficacy of our method, we study a two-dimensional prototypical problem on a matrix-void/inclusion system as a proof of concept.

为了证明我们的方法的有效性，我们研究了一个关于基质-空隙/包含系统的二维原型问题作为概念的证明。

A square-shaped matrix material contains a void/inclusion with unknown geometry.

方形基质材料包含几何结构未知的空隙/夹杂物。

To characterize the location, size, and shape of the void/inclusion, we apply loading P0 on the matrix boundary and monitor the displacement response on the measurement points at the matrix boundary under such loading.

为了表征空隙/夹杂物的位置、尺寸和形状，我们在基体边界上施加载荷P0，并在这种载荷下监测基体边界上测量点的位移响应。

We expect the PINN to inversely characterize the geometry of the void/inclusion according to the displacement data.

我们希望PINN能够根据位移数据，反过来描述空隙/夹杂物的几何特征。

重点词汇

demonstrate证明；证实；参加游行；演示；显示；表露；表现

two-dimensional二维的；平面的；没有深度的；肤浅的

prototypical原型的

void无效的；空的；空无所有的；缺门的；无用的；没有的；缺乏的；空缺；真空；空处；空位；缺门；空虚；空白；孔隙；宣告…无效；排放；排泄；倒空；疏空

proof证据；证物；证词；证言；试印品；标准酒精度；证明；验证；证实；能抵挡… 的；能耐…的；可防…的；试印的；使…防水；印…校样；试印；加水使（酵母）活化；校对；揉；发酵

unknown不知道的；不熟悉的；无名的；未被公众承认的；未知之人；未知数；未知量；未知元；未知的事物

geometry几何学；几何形状；几何构造；几何

characterize描述…的特性；形容…的特色；成为… 的特征；是…的典型

the matrix《黑客帝国》（电影名）

inversely相反地；倒转地

To test the performance of our method with various parametric assessments, we build a set of detailed cases for this problem, including different shapes and topologies of the void and different constitutive models for describing the mechanical properties of the material.

为了用各种参数评估来测试我们方法的性能，我们为这个问题建立了一组详细的案例，包括不同形状和拓扑的空洞，以及描述材料力学性能的不同本构模型。

For the particular case of inclusion, the PINN is also required to estimate the unknown material parameter of the inclusion, through which we demonstrate the capability of our model in solving combined material and geometry identification problems.

对于夹杂的特殊情况，还需要PINN来估计夹杂的未知材料参数，通过该参数，我们证明了我们的模型在解决材料和几何识别组合问题方面的能力。

In addition to the major results shown in the main text, we report in the Supplementary Materials more systematic studies of additional cases and parametric analyses, highlighting the advantages and limitations of the method.

除了正文中显示的主要结果外，我们在补充材料中报告了对其他案例和参数分析的更系统的研究，强调了该方法的优势和局限性。

重点词汇

parametric参数的；用参数表示的；确定参数的；参量放大的；参量改频的

shapes形状；身材；模型；外形；（shape的复数）；塑造；使符合；决定…的形成；（shape的第三人称单数）；形成；摆好姿势

topologies拓扑

void无效的；空的；空无所有的；缺门的；无用的；没有的；缺乏的；空缺；真空；空处；空位；缺门；空虚；空白；孔隙；宣告…无效；排放；排泄；倒空；疏空

constitutive有建立权的；有设立权的；有制定权的；有任命权的；组成的；构成的；成分的；基本的；本质的

describing形容；描述；把…称为；做…运动；画出…的形状；形成…样的形状；描绘；描写；作图；周转；(describe的现在分词)

particular case特殊情况；个别情况；特殊案例

geometry几何学；几何形状；几何构造；几何

in the main大体上, 基本上

highlighting强调；突出；使突出；照亮；（highlight的现在分词）；（文本的）重点；（皮肤或头发）的高光部分

FIG. 1. General setup of the prototypical problem on geometry and material identification in this study.

We consider a plane-strain problem in the X1-X2 plane about a square-shaped matrix specimen Ωm with a void/inclusion Ωi. Displacements are measured on the outer boundary of the matrix when loading P0 is applied. The goal is to characterize the unknown geometry of the internal void/inclusion according to the measurement data. For the case of inclusion, material properties of the inclusion are also characterized.

图1：本研究中几何和材料识别原型问题的一般设置。

我们考虑一个平面应变问题，在X1-X2平面上，关于一个方形基体试样ωm，带有一个空洞/夹杂物ωI。当施加载荷P0时，在基体的外边界上测量位移。目标是根据测量数据描述内部空隙/夹杂物的未知几何形状。对于夹杂物的情况，还描述了夹杂物的材料特性。