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In this paper, we develop a space-time upscaling framework that can be usedfor many challenging porous media applications without scale separation andhigh contrast. Our main focus is on nonlinear differential equations withmultiscale coefficients. The framework is built on nonlinear nonlocalmulti-continuum upscaling concept and significantly extends the results in theproceeding paper.  Our approach starts with a coarse space-time partition and identifies testfunctions for each partition, which plays a role of multi-continua. The testfunctions are defined via optimization and play a crucial role in nonlinearupscaling. In the second stage, we solve nonlinear local problems inoversampled regions with some constraints defined via test functions. Theselocal solutions define a nonlinear map from macroscopic variables determinedwith the help of test functions to the fine-grid fields. This map can bethought as a downscaled map from macroscopic variables to the fine-gridsolution. In the final stage, we seek macroscopic variables in the entiredomain such that the downscaled field solves the global problem in a weak sensedefined using the test functions. We present an analysis of our approach for anexample nonlinear problem.  Our unified framework plays an important role in designing various upscaledmethods. Because local problems are directly related to the fine-grid problems,it simplifies the process of finding local solutions with appropriateconstraints. Using machine learning (ML), we identify the complex map frommacroscopic variables to fine-grid solution. We present numerical results forseveral porous media applications, including two-phase flow and transport.

Space-Time Nonlinear Upscaling Framework Using Non-local Multi-continuum Approach