Machine learning and domain decomposition methods -- a survey

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning and various industrial domains, especially in computational science and engineering. In the present survey, several promising avenues of research will be examined which focus on the combination of machine learning (ML) and domain decomposition methods (DDMs). The aim of this survey is to provide an overview of existing work within this field and to structure it into domain decomposition for machine learning and machine learning-enhanced domain decomposition, including: domain decomposition for classical machine learning, domain decomposition to accelerate the training of physics-aware neural networks, machine learning to enhance the convergence properties or computational efficiency of DDMs, and machine learning as a discretization method in a DDM for the solution of PDEs. In each of these fields, we summarize existing work and key advances within a common framework and, finally, disuss ongoing challenges and opportunities for future research.

Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks

In recent years, the concept of introducing physics to machine learning has become widely popular. Most physics-inclusive ML-techniques however are still limited to a single geometry or a set of parametrizable geometries. Thus, there remains the need to train a new model for a new geometry, even if it is only slightly modified. With this work we introduce a technique with which it is possible to learn approximate solutions to the steady-state Navier--Stokes equations in varying geometries without the need of parametrization. This technique is based on a combination of a U-Net-like CNN and well established discretization methods from the field of the finite difference method.The results of our physics-aware CNN are compared to a state-of-the-art data-based approach. Additionally, it is also shown how our approach performs when combined with the data-based approach.

A Domain Decomposition-Based CNN-DNN Architecture for Model Parallel Training Applied to Image Recognition Problems

Deep neural networks (DNNs) and, in particular, convolutional neural networks (CNNs) have brought significant advances in a wide range of modern computer application problems. However, the increasing availability of large amounts of datasets as well as the increasing available computational power of modern computers lead to a steady growth in the complexity and size of DNN and CNN models, and thus, to longer training times. Hence, various methods and attempts have been developed to accelerate and parallelize the training of complex network architectures. In this work, a novel CNN-DNN architecture is proposed that naturally supports a model parallel training strategy and that is loosely inspired by two-level domain decomposition methods (DDM). First, local CNN models, that is, subnetworks, are defined that operate on overlapping or nonoverlapping parts of the input data, for example, sub-images. The subnetworks can be trained completely in parallel. Each subnetwork outputs a local decision for the given machine learning problem which is exclusively based on the respective local input data. Subsequently, an additional DNN model is trained which evaluates the local decisions of the local subnetworks and generates a final, global decision. With respect to the analogy to DDM, the DNN can be interpreted as a coarse problem and hence, the new approach can be interpreted as a two-level domain decomposition. In this paper, solely image classification problems using CNNs are considered. Experimental results for different 2D image classification problems are provided as well as a face recognition problem, and a classification problem for 3D computer tomography (CT) scans. The results show that the proposed approach can significantly accelerate the required training time compared to the global model and, additionally, can also help to improve the accuracy of the underlying classification problem.