We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge-Kutta (RK) integrator architecture. We show that we can learn high-order integrators for targeted families of differential equations without the need for computing integrator coefficients by hand. Moreover, we demonstrate that in certain cases we can obtain superior performance to classical RK methods. This can be attributed to certain properties of the ODE families being identified and exploited by the approach. Overall, this work demonstrates an effective learning-based approach to the design of algorithms for the numerical solution of differential equations. This can be readily extended to other numerical tasks.
翻译:我们研究科学计算的数字算法的学习,这种算法将数学驱动的手工设计的一般算法结构设计与数据驱动的适应特定任务类别结合起来。这与典型的数字分析方法不同,典型的数字分析方法通常不具有这种以学习为基础的适应特征。作为一个案例研究,我们开发了一种机器学习方法,根据龙格-库塔(RK)综合体结构,自动学习以普通差分方程形式解决初始价值问题的有效解决办法。我们显示,我们可以为差异方程的目标家庭学习高阶集成器,而不需要用手计算混集器系数。此外,我们证明,在某些情况下,我们可以取得优于传统RK方法的绩效。这可以归因于正在被这种方法确定和利用的ODE家庭的某些特性。总体而言,这项工作表明,在设计差异方程的数值解决方案的算法时,一种有效的以学习为基础的方法。这很容易推广到其他数字任务。