Physics-driven machine learning model on temperature and time-dependent deformation in lithium metal and its finite element implementation
锂金属随温度和时间变化变形的物理驱动机器学习模型及其有限元实现

\author{ \作者{
Jici Wen , Qingrong Zou , Yujie Wei
文继慈 , 邹庆荣 , 魏玉洁

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
北京信息科技大学应用科学学院，中国北京 100192
School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
中国科学院大学工程科学学院，中国北京 100049
}

Physics-driven machine learning
物理驱动的机器学习
Lithium-metal anode 锂金属阳极
Creep 蠕变
Finite-element analysis 有限元分析
Constitutive model 构成模型

When cataloguing deformation mechanisms and formulate physically sound and faithful constitutive laws for temperature and rate-sensitive deformation in solids, we ought to have deep understandings about the underlying kinetics and thermodynamics of deformation processes occurring in a huge span of spatial and temporal scales (Argon, 1975; Frost and Ashby, 1982; Meyers et al., 1999; Huang et al., 2018). The lack of systematic experimental data at different temperature, stress, and strain rate often leads to an incomplete understanding about the underlying multi-scale deformation, and consequentially render difficulties for the development of predictive constitutive models with high fidelity. The plasticity and creep deformation in solids, for example, are intertwined with each other and are in general a function of temperature, stress and deformation rate (Johnson and Cook, 1985). A highly predictive constitutive model, which may be contingent on limited experimental observations on the mechanical responses at different temperature and strain rate, is of compelling need for a variety of materials in engineering practice.
在编目形变机理并为固体中的温度和速率敏感形变制定物理上可靠和忠实的构效定律时，我们应该深入了解在巨大的空间和时间尺度跨度中发生的形变过程的基本动力学和热力学（Argon，1975；Frost 和 Ashby，1982；Meyers 等人，1999；Huang 等人，2018）。由于缺乏不同温度、应力和应变率下的系统实验数据，人们往往无法全面了解多尺度变形的基本原理，从而难以开发出高保真的预测构成模型。例如，固体的塑性变形和蠕变变形相互交织，通常是温度、应力和变形率的函数（Johnson 和 Cook，1985 年）。工程实践中的各种材料都迫切需要一个高度预测性的构成模型，该模型可能取决于对不同温度和应变速率下机械响应的有限实验观察。

Li-metal is such a typical example where temperature, stress and deformation rate are involved during its engineering service.
锂金属就是这样一个典型的例子，在其工程服务过程中涉及到温度、应力和变形率。

Fig. 1. The conventional vs the PD-ML based constitutive modelling for Li-metal. (a) Stress-strain curves at different temperature and at a constant strain rate , and (b) those at different strain rate at a constant temperature 298 K (experimental data from LePage et al., 2019, replotted). (c) and (d), modelling strategies: (c) A conventional method with explicit formula for plastic flow as a function of stress and temperature. (d) The PD-ML approach, where plastic flow is learned from the numerical algorithm.
图 1.传统与基于 PD-ML 的锂金属构成模型对比。(a) 不同温度和恒定应变率 下的应力-应变曲线，以及 (b) 恒温 298 K 下不同应变率下的应力-应变曲线（实验数据来自 LePage 等人，2019 年，重新绘制）。(c)和(d)，建模策略：(c) 传统方法，带有作为应力和温度函数的塑性流动显式公式。(d) PD-ML 方法，从数值算法中学习塑性流动。

Because of its highest theoretical specific capacity ( ), lowest density, and most negative potential (about -3.04 V) (Lin et al., 2017; Wood et al., 2017), Li-metal as the preferred anode is used in many new types of battery including the Li-S battery, the Li-Air battery, and the solid-state battery. However, Li-metal anode gives rise to a series of problems in Li-metal batteries (LMBs), including interfacial contact and resistance (Zhang et al., 2020; Chen et al., 2021), dendrite growth (Xu et al., 2014; Porz et al., 2017; Barai et al., 2017; Wang et al., 2020), mossy and dead Li (Chen et al., 2017), and so on. It is now well known that plastic and creep properties of Li-metal anode have great impact on the performance of the solid-state LMBs by influencing the pressure-dependent interfacial contact and resistance (Krauskopf et al., 2019). Mechanical failures of the Li-metal anode also degrade the electrochemical performance of LMBs (Kozen et al., 2017).
由于锂金属具有最高的理论比容量（ ）、最低的密度和最负的电位（约-3.04 V）（Lin 等人，2017 年；Wood 等人，2017 年），锂金属作为首选阳极被用于许多新型电池中，包括锂-S 电池、锂-空气电池和固态电池。然而，锂金属负极在锂金属电池（LMB）中引发了一系列问题，包括界面接触和电阻（Zhang 等人，2020；Chen 等人，2021）、枝晶生长（Xu 等人，2014；Porz 等人，2017；Barai 等人，2017；Wang 等人，2020）、苔藓和死锂（Chen 等人，2017）等。众所周知，锂金属阳极的塑性和蠕变特性会影响压力相关的界面接触和电阻，从而对固态 LMB 的性能产生重大影响（Krauskopf 等人，2019 年）。锂金属阳极的机械故障也会降低 LMB 的电化学性能（Kozen 等人，2017 年）。

Many related investigations have been done to shed light on the reliability of Li-metal anode due to plasticity and creep at different temperature and strain rate. Wang et al. measured the elastic-visco-plastic mechanical properties of Li-metal by nanoindentation tests (Wang and Cheng, 2017; Herbert, et al., 2018a; 2018b). Masias et al. (2019) characterized stress-strain response and creep behavior of Li-metal in tension and compression. Xu et al. (2017) and Fincher et al. (2020) explored the influence of size and strain-rate on plasticity of Li-metal. Its dependence on both strain rate and temperature was further investigated comprehensively by LePage et al. (2019). Narayan and Anand (2018) formulated a large deformation isotropic elastic-viscoplastic constitutive model for Li-metal. Those experimental investigations and modelling deepen our understanding on the mechanical response of Li-metal anode in variant conditions. There however remain challenges to construct a complete deformation map of Li-metal subject to different temperature and strain rate, or creep at different stress and temperature.
为了揭示锂金属阳极在不同温度和应变速率下因塑性和蠕变而产生的可靠性，人们进行了许多相关研究。Wang 等人通过纳米压痕试验测量了锂金属的弹塑性力学性能（Wang 和 Cheng，2017；Herbert 等人，2018a；2018b）。Masias 等人（2019）描述了 Li 金属在拉伸和压缩过程中的应力应变响应和蠕变行为。Xu 等人（2017 年）和 Fincher 等人（2020 年）探讨了尺寸和应变速率对锂金属塑性的影响。LePage 等人（2019 年）进一步全面研究了其对应变率和温度的依赖性。Narayan 和 Anand（2018 年）制定了锂金属的大变形各向同性弹性-粘弹性构成模型。这些实验研究和建模加深了我们对锂金属阳极在不同条件下机械响应的理解。然而，要构建锂金属在不同温度和应变率条件下的完整变形图，或在不同应力和温度条件下的蠕变图，仍然存在挑战。

Recent rapid development of the ML method in engineering problems where patterns or scientific principles may be extracted from big data or from strong nonlinear problems. For example, ML method, fed with data from experiments, can be adopted to describe the constitutive behavior of materials. Mozaffar et al. (2019) and Gorji et al. (2020) offers an effective method to predict path-dependent plasticity through deep learning. ML has also been used for new material design (Shi et al., 2019; Bessa et al., 2017; Curtarolo et al., 2013), structural optimization (Liu et al., 2020), property prediction (Kozuch et al., 2018; Hsu et al., 2020), multiscale modeling (Karapiperis et al., 2021), and its development and applications in special domains are rapidly growing (Li et al., 2018).
近来，从大数据或强非线性问题中提取模式或科学原理的 ML 方法在工程问题中得到迅速发展。例如，ML 方法可以利用实验数据来描述材料的构成行为。Mozaffar 等人（2019 年）和 Gorji 等人（2020 年）提供了一种通过深度学习预测路径依赖塑性的有效方法。ML 还被用于新材料设计（Shi 等人，2019；Bessa 等人，2017；Curtarolo 等人，2013）、结构优化（Liu 等人，2020）、性能预测（Kozuch 等人，2018；Hsu 等人，2020）、多尺度建模（Karapiperis 等人，2021），其在特殊领域的发展和应用也在快速增长（Li 等人，2018）。

In the paper, we introduce a new, robust, and accurate constitutive model by combining physical laws with orchestrated ML algorithm for temperature-, stress-, and rate-dependent deformation in Li-metal. We organize the paper as follows. In Section 2, the PD-
在本文中，我们将物理定律与协调 ML 算法相结合，针对锂金属中与温度、应力和速率有关的变形，介绍了一种新的、稳健且精确的构成模型。本文结构如下。在第 2 节中，PD-

Table 1 表 1
The material parameters of Li-metal, from (Krauskopf et al., 2019; LePage et al., 2019).
锂金属的材料参数，摘自（Krauskopf 等人，2019 年；LePage 等人，2019 年）。

Fig. 2. Predictability of the conventional model on the response of Li-metal under uniaxial tension. (a) The temperature dependence of curves at a constant strain rate , predictions from the conventional model vs. experiments; (b) Rate effects predicted by the conventional model vs. experiments
图 2.传统模型对单轴拉伸下锂金属 响应的可预测性。(a) 在恒定应变速率 条件下 曲线的温度依赖性，传统模型的预测与实验的对比；(b) 传统模型预测的速率效应与实验的对比 .

ML model of Li-metal and its numerical algorithm is introduced. In Section 3, we verify the PD-ML model using experimental results and demonstrate its robustness and accuracy. The PD-ML model is then implemented in the commercial FE package ABAQUS (DS Simulia Corp., 2020) as a user-material subroutine (UMAT), and we further explore the applicability of the PD-ML-FEM algorithm for the deformation of Li-metal in Section 4. We conclude in Section 5 with final remarks and discussions.
第 2 节介绍了锂金属的 ML 模型及其数值算法。在第 3 节中，我们利用实验结果验证了 PD-ML 模型，并证明了其稳健性和准确性。第 4 节中，我们进一步探讨了 PD-ML-FEM 算法在锂金属变形中的适用性。最后，我们在第 5 节中进行了总结和讨论。

To construct a faithful constitutive model which satisfying the thermodynamic principles and conservation laws, we often need systematic experimental observations to shed light on the underlying deformation process, to reveal the dependence of all variates, to calibrate unknown constants in the equations, and to validate the predictability of the constitutive model. As shown in Fig. 1, with experimental data given in Figs. 1a and b (LePage et al., 2019), we may either adopt a conventional modelling approach or a PD-ML algorithm, as to be explained next.
为了构建一个符合热力学原理和守恒定律的忠实的结构模型，我们通常需要系统的实验观测来揭示基本的变形过程，揭示所有变量的依赖关系，校准方程中的未知常数，并验证结构模型的可预测性。如图 1 所示，图 1a 和 b 中给出了实验数据（LePage et al.

For the conventional constitutive model on Li-metal, the dependence of plastic flow on stress and temperature is known as a prior. As seen in Fig. 1a, the stress-strain curves of the bulk Li-metal under uniaxial tension exhibit typical elastic-plastic response of polycrystalline metals, yielding and irrecoverable plastic deformation after the initial elasticity. The isotropic linear elastic response is characterized by a Young's modulus and a Poisson's ratio . After yielding, the Li-metal may sustain a large amount of plastic deformation with tensile failure-strain up to (Hull and Rosenberg, 1959). Li-metal has a low melting temperature ( 453 K ) and low activation energy for self-diffusion ( ) (Messer and Noack, 1975; Hao et al., 2018). The dominant mechanism accounting for the plasticity is due to the coble creep at its working temperature for electric vehicle batteries, in the range of 233 to 323 K (Chen et al., 2020; Wang et al., 2020; Zhang, 2011). It therefore leads to a highly rate-sensitive deformation, as seen in Fig. 1b.
在锂金属的传统构成模型中，塑性流动与应力和温度的关系是已知的先验关系。如图 1a 所示，块状锂金属在单轴拉伸下的应力-应变曲线表现出多晶金属的典型弹塑性响应，即在初始弹性之后出现屈服和不可恢复的塑性变形。各向同性线性弹性响应的特征是杨氏模量 和泊松比 。屈服后，锂金属可能会产生大量塑性变形，拉伸破坏应变可达 （Hull 和 Rosenberg，1959 年）。锂金属具有较低的熔化温度（453 K）和较低的自扩散活化能（ ）（Messer 和 Noack，1975 年；Hao 等人，2018 年）。塑性的主要机制是由于电动汽车电池工作温度（233 至 323 K）范围内的柯布蠕变（Chen 等人，2020 年；Wang 等人，2020 年；Zhang，2011 年）。因此，如图 1b 所示，它会导致对速率高度敏感的变形。

In the conventional approach, we use the von Mises-based visco-plastic theory with isotropic strain hardening. The Prandtl-Reuss laws is adopted for the plastic flow as
在传统方法中，我们使用了基于冯-米塞斯的粘弹性理论和各向同性应变硬化理论。塑性流动采用的普朗特-罗伊斯定律为

where is the plastic strain rate tensor, is the equivalent strain rate, is the von Mises stress, is the deviatoric part of the Cauchy stress , and is the hydrostatic pressure for being the Kronecker delta function ( if , and 0 if ). The plastic strain rate is characterized by a set of creep constants (LePage et al., 2019; Masias et al., 2019),
其中， 是塑性应变率张量， 是等效应变率， 是冯米塞斯应力， 是考奇应力的偏离部分 ， 是静水压力， 是克朗克尔三角函数（如果 为 ，如果 为0）。塑性应变率 由一组蠕变常数表征（LePage 等人，2019 年；Masias 等人，2019 年）、

Fig. 3. The integrated ML algorithm with physical assumptions. (a) Diagram to show the input to and the output from the ML model, and (b) the numerical algorithm for the constitutive model integrating PD-ML.
图 3.带有物理假设的集成 ML 算法。(a) 显示 ML 模型输入和输出的示意图；(b) 构成模型集成 PD-ML 的数值算法。

which includes a creep component , an activation energy , a kinetic constant , the corresponding reference stress , the gas constant , and the absolute temperature .
其中包括蠕变分量 、活化能 、动力学常数 、相应的参考应力 、气体常数 和绝对温度 。

We implemented the conventional constitutive model shown in Fig. 1c in the commercial finite-element (FE) software Abaqus as a user material subroutine (DS Simulia Corp., 2020). With parameters listed in Table 1, we obtain the tensile stress-strain curves at different temperature and a constant strain rate from FE calculations in Fig. 2a, and the rate-sensitivity of the mechanical response at a given temperature in Fig. 2b. We can see that the conventional power-law model fails to capture the stress hardening and rate-dependence, which, in principle, reflects the weakness of the model (see Eq. 2) in predicting the stress- and temperature-dependent creep rate. It is noted that improvements to the fitting may be possible through the use of more complicated hardening or by using series solution in the power-law creep. The hardening behavior in Li-metal may be included in the power-law creep by formulating as , where and are respectively the strain hardening modulus and exponent. We may alternatively employ series solutions and write the collective power-law flow rate as , where , are variables to be fitted. Nevertheless, determining these parameters are not trivial. In most circumstances, they are not universal and cannot be quantified uniquely. It becomes worse when experimental data is limited.
我们在商用有限元（FE）软件 Abaqus 中以用户材料子程序（DS Simulia 公司，2020 年）的形式实现了图 1c 所示的传统构成模型。利用表 1 中列出的参数，我们通过 FE 计算得到了不同温度和恒定应变速率下的拉伸应力-应变曲线（图 2a），以及给定温度下的机械响应速率敏感性（图 2b）。我们可以看到，传统的幂律模型未能捕捉到应力硬化和速率依赖性，这原则上反映了该模型（见公式 2）在预测应力和温度相关蠕变速率方面的弱点。值得注意的是，通过使用更复杂的硬化或在幂律蠕变中使用序列解，可能会改善拟合效果。通过将 表述为 ，其中 和 分别为应变硬化模量和指数，可将锂金属中的硬化行为纳入幂律蠕变。我们也可以采用系列解法，将集体幂律流速写成 ，其中 和 是待拟合的变量。然而，确定这些参数并非易事。在大多数情况下，这些参数并不通用，无法唯一量化。当实验数据有限时，情况会变得更糟。

To better capture the stress- and temperature-dependent plastic flow, we adopt a PD-ML model to supplant the regular power-law flow in Li-metal, as shown in Fig. 3. As the prior physical knowledge (physics driven, PD), those variables ( ), acquired from experimental data, are selected as inputs, where is the equivalent strain rate and is the equivalent strain, and and are the strain rate tensor and time, respectively. The outputs include the plastic strain rate and the strain hardening . In brief, the learning algorithm is a function to realize the following mapping through machine learning (ML)
为了更好地捕捉与应力和温度有关的塑性流动，我们采用了 PD-ML 模型来取代锂金属中的常规幂律流动，如图 3 所示。作为先验物理知识（物理驱动，PD），我们选择从实验数据中获取的变量 ( ) 作为输入，其中 是等效应变率， 是等效应变， 和 分别是应变率张量和时间。输出包括塑性应变率 和应变硬化 。简而言之，学习算法是通过机器学习（ML）实现以下映射的函数

Note that , in contrast to have an explicit expression in terms of its variables, represents a black-box function embedded in the PDML algorithm. The learned will be used for stress update, and is needed for the consistence condition on the yield surface. The plastic flow in Li-metal (see Eq. 3) is multi-factor influenced and highly nonlinear. Given there are only a small amount of experimental data, we choose the tree-based ML algorithm over other ML algorithms (e.g., the support vector machine, the tree-based ML algorithm, and the artificial neural network). A complete description about the learning algorithm is supplied in Appendix A.
需要注意的是， 并不是变量的明确表达式，而是 PDML 算法中嵌入的黑盒函数。学习到的 将用于应力更新，而 则需要用于屈服面上的一致性条件。锂金属中的塑性流动（见公式 3）受多因素影响，具有高度非线性。鉴于只有少量实验数据，我们选择了基于树的 ML 算法，而不是其他 ML 算法（如支持向量机、基于树的 ML 算法和人工神经网络）。关于学习算法的完整描述见附录 A。

Fig. 3 b gives the iterative process to ensure convergence. The numerical procedure composes of standard elastic predictor and visco-plastic updating. From time to , an initial predictor, by assuming the incremental strain being elastic, we obtain a trial
图 3 b 给出了确保收敛的迭代过程。数值计算过程包括标准弹性预测和粘弹性更新。从时间 到 的初始预测器，假设增量应变是弹性的，我们会得到一个试验结果

Fig. 4. Predictability of the conventional model and the PD-ML model on relation at different temperatures. (a) Traditional model vs. experiments. (b) to (d) Trained and predicted results from PD-ML model vs experiments: (b) The predicted curves at temperatures ( 348 and 398 K ) higher than those of the trained ones. (c) The predicted curves at temperatures ( 273 and 298 K ) within the trained temperature range. (d) The predicted curves at temperatures ( 198 and 248 K ) lower than those of the trained ones.
图 4.不同温度下传统模型和 PD-ML 模型对 关系的可预测性。(a) 传统模型与实验对比。(b) 至 (d) PD-ML 模型与实验的训练和预测结果：(b) 温度（348 和 398 K）下的预测曲线高于训练曲线。(c) 温度 ( 273 和 298 K ) 在训练温度范围内的预测曲线。(d) 温度 ( 198 和 248 K ) 比训练曲线低时的预测曲线。
elastic strain at current time step,
当前时间步的弹性应变 、

and update the trial stress as
并将试验应力 更新为

where is the four order elastic constant tensor. Once visco-plastic deformation involves, the implicit return-mapping PD-ML algorithm takes over to update and . With being the first guess, we proceed with an iterative process by taking the current von Mises stress , equivalent strain , temperature , and equivalent strain rate as inputs to the PD-ML and update and , see Fig. 3b for illustration. The standard associative flow rule in Eq. (1) is then used to update the plastic strain rate . Sequentially, we may write the current elastic strain and stress tensor as
其中 是四阶弹性常数张量。一旦发生粘弹性变形，隐式返回映射 PD-ML 算法就会接手更新 和 。以 为第一猜测值，我们将当前的冯米塞斯应力 、等效应变 、温度 和等效应变率 作为 PD-ML 的输入，并更新 和 ，从而进行迭代过程，见图 3b 的说明。然后使用公式 (1) 中的标准关联流规则更新塑性应变率 。依次，我们可以将当前的弹性应变和应力张量写成

and 和

Fig. 5. Deformation map of Li-metal. (a) vs. for a wide range of temperatures. (b) A three-dimensional diagram to show the temperature , stress , and plastic flow relationship. It covers a wide temperature span from 173 to 423 K .
图 5.锂金属的变形图。(a) 各种温度下 与 的关系。(b) 显示温度 、应力 和塑性流动 关系的三维图。它涵盖了 173 至 423 K 的宽温度跨度。
respectively. The iteration continues till satisfying the following criterion. The difference of between this step and the previous one, , is infinitesimal, i.e., for being a small tolerance. With known , we may proceed to update strains at time , and before the next time increment.
分别为迭代一直进行到 满足以下条件为止。这一步与上一步 之间的 差值为无穷小，即 对于 是一个小公差。在已知 的情况下，我们可以在 和 时间更新应变，然后再进行下一次增量。

We now apply both the conventional model and the PD-ML model to the viscoplastic deformation of Li-metal, and demonstrate their predictability in comparison with available experimental data. Both temperature and strain rate factors are taken into account in the two types of models.
现在，我们将传统模型和 PD-ML 模型应用于锂金属的粘塑性变形，并通过与现有实验数据的对比证明了它们的可预测性。两种模型都考虑了温度和应变率因素。

As seen in Fig. 4, we first show the normalized plastic strain rate as a function of stress from experiments at six temperatures (Fig. 4a). It is seen that increases slowly under low stress, followed by a rapid ascending at intermediate to high stress. At the late stage, plastic strain dominates and approaches to the applied strain rate , i.e., . There is a strong nonlinear dependence of and on temperature. The traditional power law model with Eq. (2) assumes intrinsically a linear relationship in the scale of vs. (curves with asterisks in Fig. 4a, and the maximum root mean square error (RMSE) is about 75%. Here RMSE , for and being the predicted value and the experimental value, respectively, and the number of data points. It fails to capture the real experimental observations.
如图 4 所示，我们首先显示了在六个温度下的实验中，归一化塑性应变率 与应力 的函数关系（图 4a）。可以看出， 在低应力下增长缓慢，随后在中高应力下迅速上升。在后期阶段，塑性应变 占主导地位，并接近应用应变率 ，即 。 和 与温度有很强的非线性关系。公式 (2) 中的传统幂律模型假定 与 与 之间存在线性关系（图 4a 中的星号曲线，最大均方根误差 (RMSE) 约为 75%）。这里的 RMSE 、 和 分别为预测值和实验值， 为数据点的数量。它无法捕捉到真实的实验观测结果。

For verifying the predictability of the PD-ML model at a series of stress and temperature range, we next divide the experimental data into two datasets: curves at four temperatures out of six are adopted for training, and the rest two are used for prediction. Here three different types of sampling are used, and the corresponding errors at different temperature are shown in Figs. 4b to d. In Fig. 4b, we choose the curves at 348 K and 398 K for prediction, which are higher than those four temperatures used for training. The extrapolated prediction leads to a prediction error about at 348 K and at 398 K . While the RSME error is significantly higher than those of the training dataset (in comparison with experimental data), the absolute errors are rather small, indicating the excellent capability for extrapolating beyond the higher end of temperature. When we choose to predict the curves at temperatures falling within those of the training dataset as shown in Fig. 4c, the predicted curves at 273 K and 298 K match very well with experimental results, and the error are about and , respectively. We show in Fig. 4 d the predictions at temperatures lower than those of the four in the training dataset. Corresponding errors are at 248 K and at 198 K . Although the predictions from the PD-ML model exhibit different level of errors based on the selection of training dataset and prediction dataset, the three characteristic sampling methods demonstrate high fidelity of the PD-ML model to capture existing experimental data. We also explored the sensitivity of the training dataset on the predictability of the learning algorithm for relationship. As shown in Fig. B1 (see Appendix B), the accuracy increases as the training dataset increases, and a minimum of stress-strain curves at three different temperature should be given. Furthermore, the model exhibits very good accuracy when generalizing to temperatures beyond the range of the training dataset, which is essential for deformation mapping as experimental data are insufficient to cover a wide range of temperature, stress and strain rate, respectively or in combination. We then used all available data at six temperatures for training, and construct the deformation map of Li-metal in Fig. 5a, where the relationship of and at a wide range of temperatures is presented.
为了验证 PD-ML 模型在一系列应力和温度范围内的可预测性，我们接下来将实验数据分为两个数据集：采用六个数据集中四个温度下的曲线进行训练，其余两个数据集用于预测。在图 4b 中，我们选择了 348 K 和 398 K 的曲线进行预测，这两个温度高于用于训练的四个温度。外推预测导致 348 K 时的预测误差约为 ，而 398 K 时的预测误差约为 。虽然 RSME 误差明显高于训练数据集的误差（与实验数据相比），但绝对误差相当小，这表明推断温度越高的能力越强。如图 4c 所示，当我们选择预测训练数据集温度范围内的 曲线时，在 273 K 和 298 K 的预测曲线与实验结果非常吻合，误差分别约为 和 。我们在图 4 d 中显示了在温度低于训练数据集中的四个温度时的预测结果。相应误差分别为 248 K 时的 和 198 K 时的 。虽然根据训练数据集和预测数据集的选择，PD-ML 模型的预测结果呈现出不同程度的误差，但三种特征采样方法表明 PD-ML 模型对现有实验数据的捕捉具有很高的保真度。我们还探讨了训练数据集对 关系学习算法可预测性的敏感性。如图所示 此外，该模型在归纳超出训练数据集范围的温度时也表现出很高的准确性，这对于变形测绘来说非常重要，因为实验数据不足以覆盖广泛的温度、应力和应变范围。此外，该模型在泛化到训练数据集范围之外的温度时也表现出很高的准确性，这对于变形绘图至关重要，因为实验数据不足以分别或组合覆盖广泛的温度、应力和应变率范围。然后，我们使用六个温度下的所有可用数据进行训练，并构建了图 5a 中的锂金属变形图，图中显示了 和 在各种温度下的关系。

Fig. 6. Predictability of the conventional model and the PD-ML model on relationship at different strain rate. (a) Traditional model vs. experiments. (b) to (d) Trained and predicted results from PD-ML model vs experiments: (b) The predicted curves at strain rate ( ) lower than those of the trained ones. (c) The predicted curves at strain rate ( within the trained rate range. (d) The predicted curves at , which is higher than those of the trained ones.
图 6.不同应变速率下传统模型和 PD-ML 模型对 关系的可预测性。(a) 传统模型与实验对比。(b) 至 (d) PD-ML 模型与实验的训练和预测结果：(b) 应变速率 ( ) 时的预测曲线低于训练曲线。(c) 应变速率 ( ) 在训练速率范围内的预测曲线。(d) 时的预测曲线，高于训练曲线。

Also, a three dimensional diagram to show the temperature , stress , and plastic flow is seen in Fig. 5b. The deformation map covers a wide temperature span from 173 K to 423 K , which covers most environmental conditions for Li-metal in engineering service.
此外，图 5b 中还显示了温度 、应力 和塑性流动 的三维图。变形图覆盖了从 173 K 到 423 K 的宽温度跨度，涵盖了锂金属在工程应用中的大多数环境条件。

Li-metal's deformation is closely subject to the time during charging and discharging process, and may be subject to dynamic loading. Rate sensitive deformation in Li-metal plays an important role on the performance of LMBs. The conventional visco-plastic constitutive model fails to capture the influence of strain rate. In the curves, the maximum RMSE is about 92%, as shown in Fig. 6a.
锂金属的变形与充放电过程中的时间密切相关，并可能受到动态负载的影响。锂金属的速率敏感变形对 LMB 的性能起着重要作用。传统的粘弹性构成模型无法捕捉应变速率的影响。如图 6a 所示，在 曲线中，最大 RMSE 约为 92%。

By adding the applied strain rate as an input to the PD-ML model (see Eq. 3), we may learn from the curves at certain loading rates and therefore predict the curves at other rates. Following the learning-prediction method we adopted for temperature dependence, we take the curves at three different strain rate as training dataset, and the rest for prediction. The strain rate of the experimental curve for prediction may be higher than, within, or lower than those rates in the training dataset. The learning vs experimental curves and the predicted one for these three sampling method are also shown in Figs. 6b to d, respectively. In Fig. 6b, we choose the curve at the strain rate for prediction. The extrapolated prediction has an RSME error about . When we choose to predict the curves at the strain rate which is within those of the training dataset ( Fig .6 c ), the error is about . We show in Fig. 6 d the predictions at strain rate higher than those of the three in the training set. The corresponding error is
通过将应用应变速率 作为 PD-ML 模型的输入（见公式 3），我们可以从特定加载速率下的 曲线中学习，从而预测其他速率下的曲线。按照温度依赖性的学习-预测方法，我们将三个不同应变速率下的 曲线作为训练数据集，其余的作为预测数据集。用于预测的实验曲线的应变率可能高于、低于或低于训练数据集中的应变率。这三种采样方法的学习与实验曲线以及预测曲线也分别如图 6b 至 d 所示。在图 6b 中，我们选择应变速率 时的曲线进行预测。外推预测的 RSME 误差约为 。当我们选择预测应变速率为 时的 曲线时（图 6 c），误差约为 。图 6 d 显示的是应变率高于训练集中三个应变率时的预测结果。相应误差为

Fig. 7. Predictability of the PD-ML model on curves under uniaxial tension. (a) Training and prediction results from PD-ML model vs. experiments at different temperature ( ; (b) Training and prediction from PD-ML model vs. experiments at different strain rate ( 298 К).
图 7.PD-ML 模型对单轴拉伸下 曲线的预测能力。(a) 不同温度 ( ) 下 PD-ML 模型与实验的训练和预测结果；(b) 不同应变率 ( 298 К) 下 PD-ML 模型与实验的训练和预测结果。

Fig. 8. Schematic diagram to show the integration of physics-driven machine learning based constitutive modelling with finite element procedures.
图 8.显示基于物理驱动的机器学习构成建模与有限元程序整合的示意图。
0.33%. Although the predictions from the PD-ML model exhibit different errors based on the selection of training dataset and prediction dataset, the three characteristic sampling methods demonstrate the high fidelity of the PD-ML model to capture existing experimental data.
0.33%.尽管根据训练数据集和预测数据集的选择，PD-ML 模型的预测结果会出现不同的误差，但这三种特征采样方法证明了 PD-ML 模型对现有实验数据的高保真度。

With the stress-strain curves reported by LePage et al. (2019), we can now examine the predictability of the PD-ML model for a variety of temperatures and strain rates. We show in Fig. 7a the curves under uniaxial tensile test from both the PD-ML model and experiments at different temperature and at a constant strain rate of . In contrast to the same prediction from the conventional model shown in Fig. 2a, we see that at all temperatures the former performs better to capture not only the nonlinear strain-hardening at small strains, but also the stress plateau at large strains. In Figs. 7 b and 2 b , predictions on the stress-strain responses at different strain rate and at constant temperature from the PD-ML model and the conventional model are shown, respectively. The conventional model fails to capture the distinct strain hardening at different strain rate; the PD-ML model leads to good agreement with experiment curves for training and those for prediction. Therefore, it is capable of predicting the stress-strain behavior of Li-metal at arbitrary temperatures and at strain rates of engineering interest.
有了 LePage 等人（2019 年）报告的应力-应变曲线，我们现在就可以检验 PD-ML 模型在各种温度和应变速率下的可预测性了。我们在图 7a 中显示了 PD-ML 模型和实验在不同温度和恒定应变率 下进行单轴拉伸试验时的 曲线。与图 2a 所示的传统模型预测结果相比，我们发现在所有温度下，前者不仅能更好地捕捉小应变时的非线性应变硬化，而且还能捕捉大应变时的应力高原。图 7 b 和图 2 b 分别显示了 PD-ML 模型和传统模型在不同应变速率和恒温 条件下的应力-应变响应预测。传统模型未能捕捉到不同应变速率下的明显应变硬化；PD-ML 模型的训练曲线和预测曲线与实验曲线十分吻合。因此，该模型能够预测锂金属在任意温度和工程学感兴趣的应变速率下的应力-应变行为。

In Section 3, we have shown that the PD-ML model is more accurate and robust than the conventional constitutive model for
在第 3 节中，我们证明了 PD-ML 模型比传统的构成模型更精确、更稳健。

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。

Wei conceived the research. Wen developed the physics-driven ML model. Wen and Zou carried out data processing and analysis. Wei and Wen write the paper. All authors read and approved the final version.
Wei 构思了这项研究。Wen 开发了物理驱动的 ML 模型。Wen 和 Zou 进行了数据处理和分析。Wei和Wen撰写论文。所有作者阅读并批准了最终版本。

The authors acknowledge supports from the NSFC Basic Science Center for 'Multiscale Problems in Nonlinear Mechanics' (No. 11988102), the National Natural Science Foundation of China (NSFC) (No. 12002343), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB22020200), and CAS Center for Excellence in Complex System Mechanics, Zou thanks the Scientific Research Foundation Project of Beijing Information Science and Technology University (No. 2025032).
作者感谢国家自然科学基金委 "非线性力学中的多尺度问题 "基础科学研究中心（编号：11988102）、国家自然科学基金委（编号：12002343）、中国科学院战略性先导科技专项（XDB22020200）和中科院复杂系统力学卓越研究中心的支持，邹晓明感谢北京信息科技大学科研基金项目（编号：2025032）的支持。

We adopt the tree-based algorithm for regression (Gradient boosting regression), which is composed of multiple regression trees to capture the nonlinear interaction relationship between the features and the target (Natekin and Knoll, 2013; Friedman et al., 2000), and its algorithm structure is summarized in Fig. 3a. We arrive with the dataset , where and refer to the input variables and its corresponding output variables, respectively, and is the total number of samples. In order to reconstruct the unknown functional dependence between and with our estimate , the initial model with a constant value is given as
我们采用基于树的回归算法（梯度提升回归），该算法由多个回归树组成，以捕捉特征与目标之间的非线性交互关系（Natekin 和 Knoll，2013 年；Friedman 等人，2000 年），其算法结构如图 3a 所示。我们得到数据集 ，其中 和 分别指输入变量及其对应的输出变量， 是样本总数。为了用我们的估计值 重建 和 之间的未知函数依赖关系，带有常数的初始模型为

which satisfies the loss function minimization. In the paper, the classic squared-error loss function is used as . We set tree base learners in the model, in which the -th tree base learner is estimated by the gradient descent procedure on the residuals as
满足损失函数最小化。本文使用经典的平方误差 损失函数 。我们在模型中设置了 个树基学习器，其中 个树基学习器 是通过对残差的梯度下降过程估计出来的，其值为

and the parameters of the -th tree base learner, , are given as
而 树基学习器 的参数为

and the multiplier are
和乘数 分别是

And finally, we update the model as
最后，我们将模型更新为

In this tree-based learning algorithm, the number of trees , the maximum depth of individual regression estimators and the minimum number of samples in a node for controlling over-fitting, and the learning rate for the impact of each tree on the final outcome, are tuned to minimize the loss function. Those hyper-parameters are optimized by using the grid-search strategy, and the experimental data are splitting into a training dataset and a prediction dataset, as we introduced in Section 3.
在这种基于树的学习算法中，树的数量 、单个回归估计器的最大深度和节点中用于控制过拟合的最小样本数，以及每棵树对最终结果影响的学习率，都是为了最小化 损失函数而调整的。这些超参数通过网格搜索策略进行优化，而实验数据则分成训练数据集和预测数据集，正如我们在第 3 节中介绍的那样。

Fig. B1 图 B1

Fig. B1. The sensitivity of training dataset on the predictability of the PD-ML model on relation at different temperature. (a) to (c) Trained and predicted results from PD-ML model vs experiments, and (d) to (f) the corresponding RMSE errors at different temperature.
图 B1.不同温度下训练数据集对 PD-ML 模型预测 关系的敏感性。(a) 至 (c) PD-ML 模型的训练和预测结果与实验结果的对比，以及 (d) 至 (f) 不同温度下相应的均方根误差。

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