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

TL;DR

This paper introduces R2N2, a recursive neural network architecture designed to learn and adapt iterative algorithms for solving various computational problems, demonstrating similarities to classical solvers.

Contribution

The paper generalizes the Runge-Kutta neural network to a recursive superstructure that modularly learns iterative algorithms, bridging neural networks and classical numerical methods.

Findings

R2N2 learns Krylov-like solvers for linear systems

It develops Newton-Krylov methods for nonlinear problems

It mimics Runge-Kutta integrators for differential equations

Abstract

Meta-learning 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 meta-learning 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 (R2N2) 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.