This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training data sets. In these settings, standard, single-domain approaches may be too inaccurate or may overfit and hence generalize poorly. Moreover, processing large-scale data sets typically requires significant memory and computing resources which can render single-domain approaches computationally prohibitive. To address these challenges, we introduce a domain decomposition formulation into the construction of a data-driven reduced model. In doing so, the basis functions used in the reduced model approximation become localized in space, which can increase the accuracy of the domain-decomposed approximation of the complex dynamics. The decomposition furthermore reduces the memory and computing requirements to process the underlying large-scale training data set. We demonstrate the effectiveness and scalability of our approach in a large-scale three-dimensional unsteady rotating detonation rocket engine simulation scenario with over $75$ million degrees of freedom and a sparse training data set. Our results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure by up to $13 \%$ and up to $5\%$ for other key quantities, such as temperature, and fuel and oxidizer mass fractions. Lastly, our approach decreases the memory requirements for processing by almost a factor of four, which in turn reduces the computing requirements as well.

翻译：本文聚焦于针对具有复杂动力学特性及稀疏训练数据集的大规模数值模拟，构建精确且具备预测能力的数据驱动降阶模型。在此类场景下，传统单域方法可能精度不足，或产生过拟合而导致泛化性能低下。此外，处理大规模数据集通常需要大量内存与计算资源，这使得单域方法在计算上难以实施。为解决上述挑战，我们提出将区域分解策略引入数据驱动降阶模型的构建过程中。通过此方式，降阶模型近似中使用的基函数可实现空间局部化，从而提升对复杂动力学过程进行区域分解近似的精度。同时，区域分解降低了处理底层大规模训练数据集所需的内存与计算需求。我们在一项超过7500万自由度且训练数据稀疏的三维非定常旋转爆震火箭发动机模拟场景中验证了所提方法的有效性及可扩展性。结果表明：相较于单域方法，区域分解版本的压力训练误差与预测误差均降低高达13%，温度、燃料与氧化剂质量分数等关键量的误差则降低高达5%。最后，本方法将数据处理的内存需求降低近四倍，进而同步减少了计算资源消耗。