Sohei Arisaka

Sohei Arisaka, Qianxiao Li

Iterative methods are ubiquitous in large-scale scientific computing applications, and a number of approaches based on meta-learning have been recently proposed to accelerate them. However, a systematic study of these approaches and how they differ from meta-learning is lacking. In this paper, we propose a framework to analyze such learning-based acceleration approaches, where one can immediately identify a departure from classical meta-learning. We show that this departure may lead to arbitrary deterioration of model performance. Based on our analysis, we introduce a novel training method for learning-based acceleration of iterative methods. Furthermore, we theoretically prove that the proposed method improves upon the existing methods, and demonstrate its significant advantage and versatility through various numerical applications.

Sohei Arisaka, Qianxiao Li

Scientific computing is an essential tool for scientific discovery and engineering design, and its computational cost is always a main concern in practice. To accelerate scientific computing, it is a promising approach to use machine learning (especially meta-learning) techniques for selecting hyperparameters of traditional numerical methods. There have been numerous proposals to this direction, but many of them require automatic-differentiable numerical methods. However, in reality, many practical applications still depend on well-established but non-automatic-differentiable legacy codes, which prevents practitioners from applying the state-of-the-art research to their own problems. To resolve this problem, we propose a non-intrusive methodology with a novel gradient estimation technique to combine machine learning and legacy numerical codes without any modification. We theoretically and numerically show the advantage of the proposed method over other baselines and present applications of accelerating established non-automatic-differentiable numerical solvers implemented in PETSc, a widely used open-source numerical software library.