  In this paper, we develop a space-time upscaling framework that can be used
for many challenging porous media applications without scale separation and
high contrast. Our main focus is on nonlinear differential equations with
multiscale coefficients. The framework is built on nonlinear nonlocal
multi-continuum upscaling concept and significantly extends the results in the
proceeding paper.
  Our approach starts with a coarse space-time partition and identifies test
functions for each partition, which plays a role of multi-continua. The test
functions are defined via optimization and play a crucial role in nonlinear
upscaling. In the second stage, we solve nonlinear local problems in
oversampled regions with some constraints defined via test functions. These
local solutions define a nonlinear map from macroscopic variables determined
with the help of test functions to the fine-grid fields. This map can be
thought as a downscaled map from macroscopic variables to the fine-grid
solution. In the final stage, we seek macroscopic variables in the entire
domain such that the downscaled field solves the global problem in a weak sense
defined using the test functions. We present an analysis of our approach for an
example nonlinear problem.
  Our unified framework plays an important role in designing various upscaled
methods. Because local problems are directly related to the fine-grid problems,
it simplifies the process of finding local solutions with appropriate
constraints. Using machine learning (ML), we identify the complex map from
macroscopic variables to fine-grid solution. We present numerical results for
several porous media applications, including two-phase flow and transport.
