Personalized Algorithm Generation: A Case Study in Learning ODE Integrators

TL;DR

This paper introduces a machine learning approach to automatically learn and optimize ODE solvers based on Runge-Kutta architecture, outperforming classical methods in specific cases by exploiting properties of targeted differential equation families.

Contribution

It presents a novel data-driven method for learning high-order ODE integrators tailored to specific problem classes, bypassing manual coefficient computation.

Findings

Learned integrators outperform classical RK methods in certain cases.

The approach automatically adapts to specific ODE families.

Demonstrates potential for extending to other numerical algorithms.

Abstract

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.