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.

中文翻译：

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使用非局部多连续体方法的时空非线性放大框架

在本文中，我们开发了一个时空放大框架，可用于许多具有挑战性的多孔介质应用，而没有尺度分离和高对比度。我们主要关注具有多尺度系数的非线性微分方程。该框架建立在非线性非局部多连续体放大概念的基础上，并显着扩展了论文中的结果。我们的方法从一个粗略的时空分区开始，并为每个分区确定测试函数，这起到了多连续体的作用。测试函数是通过优化定义的，在非线性放大中起着至关重要的作用。在第二阶段，我们使用通过测试函数定义的一些约束来解决过采样区域中的非线性局部问题。这些局部解定义了从借助测试函数确定的宏观变量到细网格域的非线性映射。该图可以被认为是从宏观变量到细网格解的缩小图。在最后阶段，我们在整个域中寻找宏观变量，以便缩小的域解决使用测试函数定义的弱意义上的全局问题。我们对示例非线性问题的方法进行了分析。我们的统一框架在设计各种升级方法方面发挥着重要作用。由于局部问题与细网格问题直接相关，因此它简化了寻找具有适当约束的局部解决方案的过程。使用机器学习 (ML)，我们识别从宏观变量到精细网格解决方案的复杂映射。