Forward uncertainty quantification (UQ) for partial differential equations is a many-query task that requires a significant number of model evaluations. The objective of this work is to mitigate the computational cost of UQ for a 3D-1D multiscale computational model of microcirculation. To this purpose, we present a deep learning enhanced multi-fidelity Monte Carlo (DL-MFMC) method that integrates the information of a multiscale full-order model (FOM) with that coming from a deep learning enhanced non-intrusive projection-based reduced order model (ROM). The latter is constructed by leveraging on proper orthogonal decomposition (POD) and mesh-informed neural networks (previously developed by the authors and co-workers), integrating diverse architectures that approximate POD coefficients while introducing fine-scale corrections for the microstructures. The DL-MFMC approach provides a robust estimator of specific quantities of interest and their associated uncertainties, with optimal management of computational resources. In particular, the computational budget is efficiently divided between training and sampling, ensuring a reliable estimation process suitably exploiting the ROM speed-up. Here, we apply the DL-MFMC technique to accelerate the estimation of biophysical quantities regarding oxygen transfer and radiotherapy outcomes. Compared to classical Monte Carlo methods, the proposed approach shows remarkable speed-ups and a substantial reduction of the overall computational cost.

翻译：偏微分方程的前向不确定性量化(UQ)是一项需要大量模型评估的多查询任务。本文旨在降低微循环三维-一维多尺度计算模型的不确定性量化计算成本。为此，我们提出了一种深度学习增强的多保真度蒙特卡洛(DL-MFMC)方法，该方法将多尺度全阶模型(FOM)的信息与深度学习增强的非侵入式投影降阶模型(ROM)的信息相结合。后者通过利用本征正交分解(POD)和网格感知神经网络(由作者及合作者先前开发)构建，集成了逼近POD系数并引入微结构细尺度修正的多种架构。DL-MFMC方法提供了目标物理量及其相关不确定性的稳健估计，并实现了计算资源的最优管理。具体而言，计算预算在训练与采样间高效分配，确保利用ROM加速特性实现可靠的估计过程。本文将DL-MFMC技术应用于加速氧传输及放疗结果相关的生物物理量估计。与经典蒙特卡洛方法相比，所提方法展现出显著加速效果，并大幅降低了总体计算成本。