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The optimal design of neural networks is a critical problem in manyapplications. Here, we investigate how dynamical systems with polynomialnonlinearities can inform the design of neural systems that seek to emulatethem. We propose a Learnability metric and its associated features to quantifythe near-equilibrium behavior of learning dynamics. Equating the Learnabilityof neural systems with equivalent parameter estimation metric of the referencesystem establishes bounds on network structure. In this way, norms from theoryprovide a good first guess for neural structure, which may then further adaptwith data. The proposed approach neither requires training nor training data.It reveals exact sizing for a class of neural networks with multiplicativenodes that mimic continuous- or discrete-time polynomial dynamics. It alsoprovides relatively tight lower size bounds for classical feed-forward networksthat is consistent with simulated assessments.

Learn Like The Pro: Norms from Theory to Size Neural Computation