Gradient-based Meta-solving and Its Applications to Iterative Methods for Solving Differential Equations

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 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 initial guesses for iterative solvers to speed up the solution of differential equations. We demonstrate the performance and versatility of our method through theoretical analysis and numerical experiments.