Online learning to accelerate nonlinear PDE solvers: applied to multiphase porous media flow

We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed method rely on four pillars: (i) dimensionless numbers as input parameters for the machine learning model, (ii) simplified numerical model (two-dimensional) for the offline training, (iii) dynamic control of a nonlinear solver tuning parameter (numerical relaxation), (iv) and online learning for real-time improvement of the machine learning model. This strategy decreases the number of nonlinear iterations by dynamically modifying a single global parameter, the relaxation factor, and by adaptively learning the attributes of each numerical model on-the-run. Furthermore, this work performs a sensitivity study in the dimensionless parameters (machine learning features), assess the efficacy of various machine learning models, demonstrate a decrease in nonlinear iterations using our method in more intricate, realistic three-dimensional models, and fully couple a machine learning model into an open-source multiphase flow simulator achieving up to 85\% reduction in computational time.

An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes

The modelling of multiphase flow in a pipe presents a significant challenge for high-resolution computational fluid dynamics (CFD) models due to the high aspect ratio (length over diameter) of the domain. In subsea applications, the pipe length can be several hundreds of kilometres versus a pipe diameter of just a few inches. In this paper, we present a new AI-based non-intrusive reduced-order model within a domain decomposition framework (AI-DDNIROM) which is capable of making predictions for domains significantly larger than the domain used in training. This is achieved by using domain decomposition; dimensionality reduction; training a neural network to make predictions for a single subdomain; and by using an iteration-by-subdomain technique to converge the solution over the whole domain. To find the low-dimensional space, we explore several types of autoencoder networks, known for their ability to compress information accurately and compactly. The performance of the autoencoders is assessed on two advection-dominated problems: flow past a cylinder and slug flow in a pipe. To make predictions in time, we exploit an adversarial network which aims to learn the distribution of the training data, in addition to learning the mapping between particular inputs and outputs. This type of network has shown the potential to produce realistic outputs. The whole framework is applied to multiphase slug flow in a horizontal pipe for which an AI-DDNIROM is trained on high-fidelity CFD simulations of a pipe of length 10 m with an aspect ratio of 13:1, and tested by simulating the flow for a pipe of length 98 m with an aspect ratio of almost 130:1. Statistics of the flows obtained from the CFD simulations are compared to those of the AI-DDNIROM predictions to demonstrate the success of our approach.