In this paper, we propose the MultiLevel Variational MultiScale (ML-VMS) method, a novel approach that seamlessly integrates a multilevel mesh strategy into the Variational Multiscale (VMS) framework. A key feature of the ML-VMS method is the use of the Convolutional Hierarchical Deep Neural Network (C-HiDeNN) as the approximation basis. The framework employs a coarse mesh throughout the domain, with localized fine meshes placed only in subdomains of high interest, such as those surrounding a source. Solutions at different resolutions are robustly coupled through the variational weak form and interface conditions. Compared to existing multilevel methods, ML-VMS (1) can couple an arbitrary number of mesh levels across different scales using variational multiscale framework; (2) allows approximating functions with arbitrary orders with linear finite element mesh due to the C-HiDeNN basis; (3) is supported by a rigorous theoretical error analysis; (4) features several tunable hyperparameters (e.g., order $p$, patch size $s$) with a systematic guide for their selection. We first show the theoretical error estimates of ML-VMS. Then through numerical examples, we demonstrate that ML-VMS with the C-HiDeNN takes less computational time than the FEM basis given comparable accuracy. Furthermore, we incorporate a space-time reduced-order model (ROM) based on C-HiDeNN-Tensor Decomposition (TD) into the ML-VMS framework. For a large-scale single-track laser powder bed fusion (LPBF) transient heat transfer problem that is equivalent to a full-order finite element model with $10^{10}$ spatial degrees of freedom (DoFs), our 3-level ML-VMS C-HiDeNN-TD achieves an approximately 5,000x speedup on a single CPU over a single-level linear FEM-TD ROM.

翻译：本文提出了多级变分多尺度（ML-VMS）方法，这是一种将多级网格策略无缝集成到变分多尺度（VMS）框架中的新方法。ML-VMS方法的一个关键特征是使用卷积层次深度神经网络（C-HiDeNN）作为近似基函数。该框架在整个计算域采用粗网格，仅在高度关注的子域（例如源周围区域）布置局部细网格。不同分辨率下的解通过变分弱形式及界面条件进行鲁棒耦合。与现有的多级方法相比，ML-VMS具有以下优势：（1）能够利用变分多尺度框架耦合任意数量、跨越不同尺度的网格层级；（2）得益于C-HiDeNN基函数，可在线性有限元网格上实现任意阶次的函数逼近；（3）具备严格的理论误差分析支撑；（4）包含多个可调超参数（例如阶数$p$、单元片尺寸$s$）并配有系统化的选取指南。我们首先给出了ML-VMS的理论误差估计。随后通过数值算例证明，在精度相当的情况下，采用C-HiDeNN的ML-VMS比采用有限元基函数的方法计算耗时更少。此外，我们将基于C-HiDeNN-张量分解（TD）的时空降阶模型（ROM）集成到ML-VMS框架中。针对一个相当于具有$10^{10}$个空间自由度（DoFs）的全阶有限元模型的大规模单道激光粉末床熔融（LPBF）瞬态传热问题，我们的三级ML-VMS C-HiDeNN-TD方法在单CPU上相比单级线性FEM-TD ROM实现了约5,000倍的加速。