A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms

SIAM Journal on Scientific Computing, vol. 46, pp. A719–A743

Abstract

Abstract. Metalearning of numerical algorithms for a given task consists of the data-driven identification and adaptation of an algorithmic structure and the associated hyperparameters. To limit the complexity of the metalearning problem, neural architectures with a certain inductive bias towards favorable algorithmic structures can, and should, be used. We generalize our previously introduced Runge–Kutta neural network to a recursively recurrent neural network superstructure for the design of customized iterative algorithms. In contrast to off-the-shelf deep learning approaches, it features a distinct division into modules for generation of information and for the subsequent assembly of this information towards a solution. Local information in the form of a subspace is generated by subordinate, inner, iterations of recurrent function evaluations starting at the current outer iterate. The update to the next outer iterate is computed as a linear combination of these evaluations, reducing the residual in this space, and constitutes the output of the network. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields iterations similar to Krylov solvers for linear equation systems, Newton–Krylov solvers for nonlinear equation systems, and Runge–Kutta integrators for ordinary differential equations. Due to its modularity, the superstructure can be readily extended with functionalities needed to represent more general classes of iterative algorithms traditionally based on Taylor series expansions.

Authors 7

1. Danimir T. Doncevic Aachen

   RWTH Aachen University · Forschungszentrum Jülich

   Affiliation as printed

   Institute of Energy and Climate Research, Energy Systems Engineering (IEK-10), Forschungszentrum Jülich GmbH, Jülich, 52425, Germany; RWTH Aachen University Aachen, 52062, Germany

2. Alexander Mitsos Aachen

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   JARA-ENERGY, Jülich, 52425, Germany; Institute of Energy and Climate Research, Energy Systems Engineering (IEK-10), Forschungszentrum Jülich GmbH, Jülich, 52425, Germany; RWTH Aachen University, Process Systems Engineering (AVT.SVT), Aachen, 52074, Germany

3. Y. P. Guo

   National University of Singapore

   Affiliation as printed

   Department of Mathematics, National University of Singapore, 117543, Singapore

4. Qianxiao Li

   National University of Singapore

   Affiliation as printed

   Department of Mathematics, National University of Singapore, 117543, Singapore

5. Felix Dietrich

   Technical University of Munich

   Affiliation as printed

   Department of Informatics, Technical University of Munich, Boltzmannstr. 3, 85748, Garching b. Munich, Germany

6. Manuel Dahmen

   Forschungszentrum Jülich

   Affiliation as printed

   Corresponding author. Institute of Energy and Climate Research, Energy Systems Engineering (IEK-10), Forschungszentrum Jülich GmbH, Jülich, 52425, Germany

7. Ioannis G. Kevrekidis

   Johns Hopkins University

   Affiliation as printed

   Corresponding author. Departments of Applied Mathematics and Statistics & Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD 21218 USA