Accelerating Legacy Numerical Solvers by Non-intrusive Gradient-based Meta-solving

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.

Accelerating numerical methods by gradient-based meta-solving

In science and engineering applications, it is often required to solve similar computational problems repeatedly. In such cases, we can utilize the data from previously solved problem instances to improve the efficiency of finding subsequent solutions. This offers a unique opportunity to combine machine learning (in particular, meta-learning) and scientific computing. To date, a variety of such domain-specific methods have been proposed in the literature, but a generic approach for designing these methods remains under-explored. In this paper, we tackle this issue by formulating a general framework to describe these problems, and propose a gradient-based algorithm to solve them in a unified way. As an illustration of this approach, we study the adaptive generation of parameters for iterative solvers to accelerate the solution of differential equations. We demonstrate the performance and versatility of our method through theoretical analysis and numerical experiments, including applications to incompressible flow simulations and an inverse problem of parameter estimation.