The weak imposition of essential boundary conditions is an integral aspect of unfitted finite element methods, where the physical boundary does not in general coincide with the computational domain. In this regard, the symmetric Nitsche's method is a powerful technique that preserves the symmetry and variational consistency of the unmodified weak formulation. The stabilization parameter in Nitsche's method plays a crucial role in the stability of the resultant formulation, whose estimation is computationally intensive and dependent on the particular cut configuration using the conventional eigenvalue-based approach. In this work, we employ as model problem the finite cell method in which the need for the generation of a boundary-conforming mesh is circumvented by embedding the physical domain in a, typically regular, background mesh. We propose a data-driven estimate based on machine learning methods for the estimation of the stabilization parameter in Nitsche's method that offers an efficient constant-complexity alternative to the eigenvalue-based approach independent of the cut configuration. It is shown, using numerical benchmarks, that the proposed method can estimate the stabilization parameter accurately and is by far more computationally efficient. The data-driven estimate can be integrated into existing numerical codes with minimal modifications and thanks to the wide adoption of accelerators such as GPUs by machine learning frameworks, can be used with virtually no extra implementation cost on GPU devices, further increasing the potential for computational gains over the conventional eigenvalue-based estimate. The proposed model is tested on both Intel CPU as well as NVIDIA GPU hardware, showing that while it is already many times more efficient on the CPU compared to the eigenvalue-based estimate, its efficiency margin is even larger on modern GPU devices.

翻译：在非拟合有限元方法中，物理边界通常不与计算域重合，因此弱施加本质边界条件是其核心组成部分。对称尼采方法作为一种强大技术，能够保持未修正弱形式的对称性与变分一致性。尼采方法中的稳定化参数对最终公式的稳定性至关重要，传统基于特征值的方法在估计该参数时计算成本高昂，且依赖于特定的切割构型。本研究以有限单元法为模型问题，通过将物理域嵌入通常规则的背景网格中，规避了生成边界贴合网格的需求。我们提出一种基于机器学习方法的数据驱动估计策略，用于确定尼采方法中的稳定化参数。该方法提供了独立于切割构型、具有恒定计算复杂度的高效替代方案，可取代传统特征值方法。数值基准测试表明，所提方法能准确估计稳定化参数，且计算效率显著提升。该数据驱动估计方案只需对现有数值代码进行最小修改即可集成，并借助机器学习框架广泛采用的GPU等加速器，在GPU设备上几乎无需额外实现成本，进一步提升了相较于传统特征值估计的计算增益潜力。所提模型在Intel CPU和NVIDIA GPU硬件上的测试结果显示：虽然在CPU上已较特征值估计方法效率提升数倍，其在现代GPU设备上的效率优势更为显著。