用于非线性放大闭合的显式物理学信息神经网络：组织中传输的情况
Explicit physics-informed neural networks for non-linear upscaling closure: the case of transport in tissues
摘要：
在这项工作中，我们使用了正式的升级和数据驱动的机器学习的组合，以显式地关闭多尺度组织中的非线性传输和反应过程。经典的有效因子模型被用来制定宏观反应动力学。我们训练一个多层感知器网络使用训练数据直接数值模拟产生的微尺度的例子。训练完毕后，该网络用于数值求解描述两个示例组织中质量传输和反应的升级（粗粒度）微分方程。该网络被描述为是明确的，在这个意义上，该网络是使用宏观浓度和浓度梯度作为特征空间的组成部分进行训练的。网络训练和解决方案的宏观传输方程计算两种不同的组织。两种组织类型（脑和肝）表现出明显不同的几何复杂性和空间尺度（细胞大小和样本大小）。放大的平均浓度的解决方案进行比较，从微尺度浓度场的后验平均的数值解。这项工作有两个特别值得注意的结果：1）我们发现，训练的网络表现出良好的泛化能力，并且尽管两种示例组织类型的尺度和几何形状显著不同，但它能够以高保真度预测真实结构组织的有效因子;以及2）该方法导致具有经由经训练的神经网络（隐式地）预测的有效性因子的放大PDE。后一个结果强调了我们在传统平均方法与使用机器学习闭合之间的有目的的联系;这与一些用于放大的机器学习方法形成对比，其中宏观尺度方程的确切形式仍然未知。
英文摘要：
In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in a multiscale tissue. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space. Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are two outcomes of this work of particular note: 1) we find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometry of the two example tissue types; and 2) the approach results in an upscaled PDE with an effectiveness factor that is predicted (implicitly) via the trained neural network. This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling where the exact form of the macroscale equation remains unknown.
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