This review article highlights state-of-the-art data-driven techniques to
discover, encode, surrogate, or emulate constitutive laws that describe the
path-independent and path-dependent response of solids. Our objective is to
provide an organized taxonomy to a large spectrum of methodologies developed in
the past decades and to discuss the benefits and drawbacks of the various
techniques for interpreting and forecasting mechanics behavior across different
scales. Distinguishing between machine-learning-based and model-free methods,
we further categorize approaches based on their interpretability and on their
learning process/type of required data, while discussing the key problems of
generalization and trustworthiness. We attempt to provide a road map of how
these can be reconciled in a data-availability-aware context. We also touch
upon relevant aspects such as data sampling techniques, design of experiments,
verification, and validation.

该综述论文重点介绍了最先进的数据驱动技术，用于发现、编码、替代或仿真描述固体无路径依赖响应的本构法则。我们旨在提供一个有组织的分类体系，介绍过去几十年中开发的各种方法的优点和缺点，并讨论在不同尺度上解释和预测力学行为的不同技术。我们将方法分为基于机器学习和无模型方法，并根据解释能力和所需数据学习过程 / 类型进一步分类，同时讨论了一般化和可信性的关键问题。我们试图提供一个数据可用性感知背景下如何解决这些问题的路线图，并涉及数据采样技术、实验设计、验证和验证等相关方面。

固体材料数据驱动本构定律综述

A review on data-driven constitutive laws for solids

Neural operators, which can act as implicit solution operators of hidden
governing equations, have recently become popular tools for learning the
responses of complex real-world physical systems. Nevertheless, most neural
operator applications have thus far been data-driven and neglect the intrinsic
preservation of fundamental physical laws in data. In this work, we introduce a
novel integral neural operator architecture called the Peridynamic Neural
Operator (PNO) that learns a nonlocal constitutive law from data. This neural
operator provides a forward model in the form of state-based peridynamics, with
objectivity and momentum balance laws automatically guaranteed. As
applications, we demonstrate the expressivity and efficacy of our model in
learning complex material behaviors from both synthetic and experimental data
sets. We show that, owing to its ability to capture complex responses, our
learned neural operator achieves improved accuracy and efficiency compared to
baseline models that use predefined constitutive laws. Moreover, by preserving
the essential physical laws within the neural network architecture, the PNO is
robust in treating noisy data. The method shows generalizability to different
domain configurations, external loadings, and discretizations.

非局部代表性学习神经算子（PNO）作为前向模型展示了学习复杂材料行为的表达能力和效力，并通过在神经网络架构中保留基本物理定律对噪声数据具有鲁棒性。

周期动力神经算子：基于数据驱动的非局部本构模型与复杂材料响应

Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model  for Complex Material Responses

Nonlinear materials are often difficult to model with classical state model
theory because they have a complex and sometimes inaccurate physical and
mathematical description or we simply do not know how to describe such
materials in terms of relations between external and internal variables. In
many disciplines, Neural Network methods have arisen as powerful tools to
identify very complex and non-linear correlations. In this work, we use the
very recently developed concept of Physically Guided Neural Networks with
Internal Variables (PGNNIV) to discover constitutive laws using a model-free
approach and training solely with measured force-displacement data. PGNNIVs
make a particular use of the physics of the problem to enforce constraints on
specific hidden layers and are able to make predictions without internal
variable data. We demonstrate that PGNNIVs are capable of predicting both
internal and external variables under unseen load scenarios, regardless of the
nature of the material considered (linear, with hardening or softening behavior
and hyperelastic), unravelling the constitutive law of the material hence
explaining its nature altogether, placing the method in what is known as
eXplainable Artificial Intelligence (XAI).

利用物理导向神经网络，仅使用测量的力位移数据进行训练，实现对非线性材料的内外部变量的预测，揭示材料的本质，从而实现可解释人工智能。

使用带有内部变量的深度物理引导神经网络预测和解释非线性材料响应

Predicting and explaining nonlinear material response using deep  Physically Guided Neural Networks with Internal Variables

Multiscale partial differential equations (PDEs) arise in various
applications, and several schemes have been developed to solve them
efficiently. Homogenization theory is a powerful methodology that eliminates
the small-scale dependence, resulting in simplified equations that are
computationally tractable. In the field of continuum mechanics, homogenization
is crucial for deriving constitutive laws that incorporate microscale physics
in order to formulate balance laws for the macroscopic quantities of interest.
However, obtaining homogenized constitutive laws is often challenging as they
do not in general have an analytic form and can exhibit phenomena not present
on the microscale. In response, data-driven learning of the constitutive law
has been proposed as appropriate for this task. However, a major challenge in
data-driven learning approaches for this problem has remained unexplored: the
impact of discontinuities and corner interfaces in the underlying material.
These discontinuities in the coefficients affect the smoothness of the
solutions of the underlying equations. Given the prevalence of discontinuous
materials in continuum mechanics applications, it is important to address the
challenge of learning in this context; in particular to develop underpinning
theory to establish the reliability of data-driven methods in this scientific
domain. The paper addresses this unexplored challenge by investigating the
learnability of homogenized constitutive laws for elliptic operators in the
presence of such complexities. Approximation theory is presented, and numerical
experiments are performed which validate the theory for the solution operator
defined by the cell-problem arising in homogenization for elliptic PDEs.

多尺度偏微分方程中的组分理论及椭圆 PDE 的合成定律的数据驱动学习方法。