Parametric data-driven reduced-order models (ROMs) that embed dependencies in a large number of input parameters are crucial for enabling many-query tasks in large-scale problems. These tasks, including design optimization, control, and uncertainty quantification, are essential for developing digital twins in real-world applications. However, standard grid-based data generation methods are computationally prohibitive due to the curse of dimensionality. This paper investigates efficient training of parametric data-driven ROMs using sparse grid interpolation with (L)-Leja points, specifically targeting scenarios with higher-dimensional input parameter spaces. (L)-Leja points are nested and exhibit slow growth, resulting in sparse grids with low cardinality in low-to-medium dimensional settings, making them ideal for large-scale, computationally expensive problems. Focusing on gyrokinetic simulations of plasma micro-instabilities in fusion experiments as a representative real-world application, we construct parametric ROMs for the full 5D gyrokinetic distribution function via optimized dynamic mode decomposition (optDMD) and sparse grids based on (L)-Leja points. We perform detailed experiments in two scenarios: First, the Cyclone Base Case benchmark assesses optDMD ROM prediction capabilities beyond training time horizons and across variations in the binormal wave number. Second, for a real-world electron-temperature-gradient-driven micro-instability simulation with six input parameters, we demonstrate that a predictive parametric optDMD ROM that is up to three orders of magnitude cheaper to evaluate can be constructed using only 28 high-fidelity gyrokinetic simulations, enabled by the use of sparse grids. In the broader context of fusion research, these results demonstrate the potential of sparse grid-based parametric ROMs to enable otherwise intractable many-query tasks.

翻译：参数化数据驱动降阶模型（ROM）通过嵌入大量输入参数的依赖性，对于实现大规模问题中的多查询任务至关重要。这些任务包括设计优化、控制和不确定性量化，是现实应用中开发数字孪生的关键。然而，由于维数灾难，标准的基于网格的数据生成方法在计算上难以实现。本文研究利用基于(L)-Leja点的稀疏网格插值高效训练参数化数据驱动ROM，特别针对输入参数空间维度较高的场景。(L)-Leja点具有嵌套性且增长缓慢，在中低维设置下能生成基数较小的稀疏网格，使其非常适合大规模、计算成本高昂的问题。以聚变实验中等离子体微观不稳定性的回旋动理学模拟作为代表性实际应用，我们通过优化动态模态分解（optDMD）和基于(L)-Leja点的稀疏网格，为完整的5D回旋动理学分布函数构建了参数化ROM。我们在两种场景下进行了详细实验：首先，通过Cyclone Base Case基准测试评估optDMD ROM在训练时间范围之外以及沿双法向波数变化时的预测能力。其次，针对具有六个输入参数的实际电子温度梯度驱动微观不稳定性模拟，我们证明仅需28个高保真回旋动理学模拟即可构建预测性参数化optDMD ROM，其评估成本降低达三个数量级，这得益于稀疏网格的应用。在更广泛的聚变研究背景下，这些结果表明基于稀疏网格的参数化ROM有望实现原本难以处理的多查询任务。