DATA-DRIVEN STRESS-STRAIN MODELING FOR GRANULAR MATERIALS THROUGH DEEP REINFORCEMENT LEARNING
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摘要: 颗粒材料的宏观力学行为受颗粒组分等材料参数, 孔隙率、配位数等状态参数的影响, 同时又具备复杂的加载路径和加载历史相关性, 建立包含多个内变量以及各变量间相互关联的颗粒材料本构模型是一个重要的科学难题. 不同于传统的基于屈服面、流动法则和硬化函数框架下的唯象本构模型, 本文基于颗粒物质力学的研究基础, 以颗粒材料平均孔隙率、细观组构参数和弹性刚度参数作为内变量, 结合深度学习方法建立以有向图表征的数据本构模型. 有向图中以不同的链接网络表示不同的内变量信息流动方向, 各个内变量间的映射关系采用循环神经网络来建立, 将各个神经网络相互组合, 形成包含不同内变量且具有不同预测能力的本构模型. 该本构模型的建立过程等价于在众多可能的内变量链接关系空间中寻找最能描述实际材料宏观应力应变行为的优化问题. 因此, 可将有向图本构模型的建立过程看作“马尔可夫决策过程”, 采用深度强化学习算法构建有向图的内变量链接组合优化过程, 具体采用AlphaGo Zero算法自动寻找最优的颗粒材料数据驱动本构模型建模路径. 研究结果表明, 采用有向图和深度强化学习算法可建立起完全依靠“数据驱动”的颗粒材料应力−应变关系. 此外, 本方法提供了一种将不同理论模型从数据角度统一起来, 且基于人工智能算法发展更优模型的研究思路, 可为相似问题的研究提供借鉴.Abstract: The macroscopic mechanical behaviours of granular materials are affected by not just material properties such as particle composition but also state parameters of granular assembly, like porosity and coordination number, etc. Meanwhile, granular materials are of complex loading path- and loading history-dependent features. Establishing a constitutive model incorporating multiple internal variables and their inherent relations for granular materials is an important scientific challenge. Different from the traditional phenomenological constitutive model based on the framework of yield surface, flow rule and hardening function, this study establishes a directed graph-based data-driven constitutive model with the average porosity, fabric tensor and equivalent elastic stiffness tensor being considered as internal variables, which are critical to the constitutive behaviour of granular materials from the perspectives of the particulate mechanics. The constitutive models containing different internal variables and having different predictive capabilities are represented by different directed graphs with various internal variables linking networks. The recurrent neural network is trained to represent the source-target mapping relationships of the information flows between internal variables. The process of establishing constitutive models is simplified as a sequence of forming graph edges with the goal of finding the optimal combination of internal variables. Therefore, the modelling of the constitutive model can be formulated as a Markov decision process and implemented by the deep reinforcement learning algorithm. Specifically, the well-known AlphaGo Zero algorithm is used to automatically discover the optimal data-driven constitutive modelling path for granular materials. Our numerical examples show that this modeling framework can produce constitutive models with higher prediction accuracy. Furthermore, this study provides a new research paradigm by integrating different theoretical models from the point of data and leveraging the algorithms in artificial intelligence to develop a superior model. The same idea can be extended to seek new insights for similar scientific problems.
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图 4 一次完整的有向图本构关系建立过程
Figure 4. A complete modeling process of generating a constitutive relationship
图 7 训练集和验证集上GRU模型的训练性能
Figure 7. Training performance of GRU architecture on training and validation data
图 8 采用强化学习算法建立的最优有向图
Figure 8. Optimal directed graph of stress-strain laws learned by deep reinforcement learning
图 9 等p加载下“数据”本构模型预测比较
Figure 9. Comparison between predictions and DEM simulation results for constant-p compression
图 11 常规三轴加载下“数据”本构模型预测比较
Figure 11. Comparison between predictions and DEM simulation results for conventional triaxial compression
图 10 等b加载下“数据”本构模型预测比较
Figure 10. Comparison between predictions and DEM simulation results for constant-b compression
图 13 有向图链接{ε → Cf; Cf → σ}的预测精度
Figure 13. Prediction performance of directed graph {ε → Cf; Cf → σ}
表 1 利用强化学习算法建立最优应力−应变关系有向图流程
Table 1. Reinforcement learning of directed graph of the stress-strain relationship
定义: 有向图配置、状态、动作、模型得分、奖励、建模规则等 1 随机初始化策略/价值网络fθ 2 创建并初始化训练集Etrain 3 for i in [1, Niter]: 4 for j in [1, Ncollect]: 5 初始化有向图链接状态 s, 初始化空的MCTS搜索树, 令探索系数τ = 1 6 while True: 7 依据建模规则进行NMCTS次蒙特卡罗树搜索获得 动作策略π(s, ∙), 根据策略选择激活有向边a, 有向 图从当前状态s将转移到新状态$ s' $ 8 if $ s' $是最终状态 then 9 计算应力−应变有向图得分, 并根据有向图得分计 算奖励值r Break 10 将数据[s, a, π(s, ∙), r]加入训练集Etrain 11 利用Etrain训练策略/价值网络fθ 12 探索系数设为τ = 0.01, 利用训练后的策略/价值网络 fθ 进行一次有向图建立过程, 获得最终的应力−应变有向图 13 计算完成 
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