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
