  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.
