This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate $\textit{dynamics}$ for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ -- with negligible errors.

翻译：本文提出了一种在非线性流形上对物质点法进行模型降阶的方法。我们的技术通过使用隐式神经表示来近似变形映射，将变形轨迹限制在低维流形上，从而近似运动学。通过显式近似变形映射，其时空梯度——特别是变形梯度和速度——可以通过解析微分计算。与构建线性或非线性流形来近似表征给定空间离散化的（有限数量）自由度的典型模型降阶技术不同，隐式神经表示的使用使所提方法能够近似连续变形映射。这使得运动学近似与离散化无关。因此，该技术支持在在线降阶模型模拟过程中进行动态离散化（包括分辨率变化）。为了生成广义坐标下的动力学，我们提出了一系列投影技术。在每个时间步，这些技术：（1）计算求积点处的全空间运动学，（2）计算“样本”物质点子集的全空间动力学，以及（3）通过分别将更新后的全空间位置和速度投影到低维流形和切空间上，计算降阶空间动力学。我们通过超降阶实现了显著的计算加速，确保所有三个步骤仅在问题空间域的一小部分上执行。包含数百万个物质点的大规模数值示例说明了该方法能够获得一个数量级的计算成本节省——甚至实现实时模拟——且误差可忽略不计。