Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment.

计算物理学中的许多应用涉及微观结构的近似问题，其特征在于数据中的多个空间尺度。然而，由于需要捕捉小尺度的精细细节，这些数值解决方案的计算成本通常很高。因此，对于多查询应用程序（例如具有多个尺度相关特征的参数化系统）来说，模拟此类现象变得难以承受。传统的基于投影的降阶模型 (ROM) 无法解决这些问题，即使对于工程应用中常见的二阶椭圆偏微分方程也是如此。为了解决这个问题，我们提出了一种替代的非侵入式策略来构建 ROM，它将经典的适当正交分解 (POD) 与合适的神经网络 (NN) 模型结合起来以考虑小规模。具体来说，我们采用稀疏网格通知神经网络（MINN），它同时处理解决方案和模型参数中的空间依赖性。我们评估该策略在基准问题上的性能，然后将其应用于近似现实生活中的问题，该问题涉及微循环对组织微环境运输现象的影响。