数据驱动的近周连续介质模型及其在分子动力学中的应用
A data-driven peridynamic continuum model for upscaling molecular dynamics
摘要：
非局部模型，包括周波模型，经常使用在定义中嵌入长度尺度的积分算子。然而，这些算子中的被积函数很难从通常可用于给定物理系统的数据中定义，例如实验室机械性能测试。相比之下，分子动力学（MD）不需要这些被积函数，但它在长度和时间尺度上受到计算限制。为了结合这两种方法的优势，并获得一个粗粒度的，均匀的连续模型，有效地和准确地捕捉材料的行为，我们提出了一个学习框架，从MD数据中提取一个最佳的线性周向固体（LPS）模型作为MD位移的替代。为了最大限度地提高学习模型的准确性，我们允许周波影响函数部分为负，同时保持所得模型的适定性。为了实现这一目标，我们提供了足够的适定性条件与符号变化的影响函数的离散化LPS模型，并开发了一个约束优化算法，最大限度地减少方程残差，同时执行这样的可解性条件。这个框架保证了生成的模型在数学上是适定性的，物理上是一致的，并且它可以很好地推广到与训练过程中使用的设置不同的设置。我们说明了所提出的方法的有效性与单层石墨烯的几个数值测试。我们的二维测试表明，所提出的算法的鲁棒性验证数据集，包括热噪声，不同的域形状和外部负载，和离散化，大大不同于用于训练。
英文摘要：
Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials' behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.
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