Linear reduced-order modeling (ROM) simplifies complex simulations by approximating the behavior of a system using a simplified kinematic representation. Typically, ROM is trained on input simulations created with a specific spatial discretization, and then serves to accelerate simulations with the same discretization. This discretization-dependence is restrictive. Becoming independent of a specific discretization would provide flexibility to mix and match mesh resolutions, connectivity, and type (tetrahedral, hexahedral) in training data; to accelerate simulations with novel discretizations unseen during training; and to accelerate adaptive simulations that temporally or parametrically change the discretization. We present a flexible, discretization-independent approach to reduced-order modeling. Like traditional ROM, we represent the configuration as a linear combination of displacement fields. Unlike traditional ROM, our displacement fields are continuous maps from every point on the reference domain to a corresponding displacement vector; these maps are represented as implicit neural fields. With linear continuous ROM (LiCROM), our training set can include multiple geometries undergoing multiple loading conditions, independent of their discretization. This opens the door to novel applications of reduced order modeling. We can now accelerate simulations that modify the geometry at runtime, for instance via cutting, hole punching, and even swapping the entire mesh. We can also accelerate simulations of geometries unseen during training. We demonstrate one-shot generalization, training on a single geometry and subsequently simulating various unseen geometries.

翻译：线性降阶建模通过简化运动学表示来近似系统行为，从而简化复杂仿真。传统降阶模型通常基于特定空间离散化的输入仿真进行训练，并加速相同离散化条件下的仿真过程。这种对离散化的依赖性具有局限性。摆脱对特定离散化的依赖，可在训练数据中灵活融合不同网格分辨率、连接方式和类型（四面体、六面体）；加速训练中未见过的新型离散化仿真；以及加速随参数或时间改变离散化的自适应仿真。本文提出一种灵活、与离散化无关的降阶建模方法。与传统降阶模型类似，我们将构型表示为位移场的线性组合；但与传统方法不同的是，我们的位移场是从参考域内每一点到对应位移向量的连续映射，这些映射通过隐式神经场表征。基于线性连续降阶模型（LiCROM），训练集可包含多种几何体在多种载荷条件下的响应，且不依赖其离散化方式。这为降阶建模开辟了全新应用场景：我们可加速运行时修改几何体的仿真（如切割、打孔乃至完全替换网格），还能加速训练中未见几何体的仿真。实验证明，LiCROM可实现单样本泛化——仅基于单一几何体训练即可对各种未见几何体进行仿真。