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

## TL;DR

This paper introduces a space-time nonlinear upscaling framework for porous media that handles multiscale coefficients without scale separation, utilizing nonlocal multi-continuum concepts and machine learning for efficient solutions.

## Contribution

The paper extends previous results by developing a nonlinear nonlocal multi-continuum upscaling method applicable to complex porous media problems without scale separation.

## Key findings

- Effective upscaling for nonlinear porous media problems.
- Use of machine learning to identify complex solution maps.
- Numerical validation on two-phase flow and transport applications.

## Abstract

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.