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Meta-learning of numerical algorithms for a given task consists of thedata-driven identification and adaptation of an algorithmic structure and theassociated hyperparameters. To limit the complexity of the meta-learningproblem, neural architectures with a certain inductive bias towards favorablealgorithmic structures can, and should, be used. We generalize our previouslyintroduced Runge-Kutta neural network to a recursively recurrent neural network(R2N2) superstructure for the design of customized iterative algorithms. Incontrast to off-the-shelf deep learning approaches, it features a distinctdivision into modules for generation of information and for the subsequentassembly of this information towards a solution. Local information in the formof a subspace is generated by subordinate, inner, iterations of recurrentfunction evaluations starting at the current outer iterate. The update to thenext 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 theproposed superstructure on input/output data of various computational problemclasses yields iterations similar to Krylov solvers for linear equationsystems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kuttaintegrators for ordinary differential equations. Due to its modularity, thesuperstructure can be readily extended with functionalities needed to representmore general classes of iterative algorithms traditionally based on Taylorseries expansions.

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