We use a space-time discretization based on physics informed deep learning (PIDL) to approximate solutions of a class of rate-dependent strain gradient plasticity models. The differential equation governing the plastic flow, the so-called microforce balance for this class of yield-free plasticity models, is very stiff, often leading to numerical corruption and a consequent lack of accuracy or convergence by finite element (FE) methods. Indeed, setting up the discretized framework, especially with an elaborate meshing around the propagating plastic bands whose locations are often unknown a-priori, also scales up the computational effort significantly. Taking inspiration from physics informed neural networks, we modify the loss function of a PIDL model in several novel ways to account for the balance laws, either through energetics or via the resulting PDEs once a variational scheme is applied, and the constitutive equations. The initial and the boundary conditions may either be imposed strictly by encoding them within the PIDL architecture, or enforced weakly as a part of the loss function. The flexibility in the implementation of a PIDL technique often makes for its ready interface with powerful optimization schemes, and this in turn provides for many possibilities in posing the problem. We have used freely available open-source libraries that perform fast, parallel computations on GPUs. Using numerical illustrations, we demonstrate how PIDL methods could address the computational challenges posed by strain gradient plasticity models. Also, PIDL methods offer abundant potentialities, vis-\'a-vis a somewhat straitjacketed and poorer approximant of FE methods, in customizing the formulation as per the problem objective.

翻译：本文采用基于物理信息深度学习（PIDL）的时空离散化方法，近似求解一类率相关应变梯度塑性模型。控制塑性流动的微分方程——即这类无屈服塑性模型中的微力平衡方程——具有高度刚性，常导致有限元（FE）方法出现数值失真及随之而来的精度不足或收敛困难。实际上，建立离散化框架（尤其是在传播的塑性带周围进行精细网格划分，而这些塑性带的位置通常先验未知）会显著增加计算负担。受物理信息神经网络的启发，我们通过多种新颖方式改进PIDL模型的损失函数，以兼顾平衡定律（无论是通过能量描述，还是在应用变分格式后通过所得偏微分方程体现）和本构方程。初始条件与边界条件可通过将其编码至PIDL架构中严格施加，或作为损失函数的一部分进行弱约束执行。PIDL技术在实现上的灵活性使其易于与强大的优化方案对接，这进而为问题表述提供了多种可能性。我们使用了可免费获取的开源库，这些库能在GPU上执行快速并行计算。通过数值算例，我们展示了PIDL方法如何应对应变梯度塑性模型带来的计算挑战。相较于有限元方法相对受限且近似能力较弱的特性，PIDL方法在根据问题目标定制公式体系方面展现出巨大潜力。