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Neural Oscillators are Universal

by S. Lanthaler and T. K. Rusch and S. Mishra

(Report number 2023-20)

Abstract
Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.

Keywords: Neural Oscillators, Neural ODEs, Universal Approximation, Deep Learning, Hamiltonian systems

@Techreport{LRM23_1057,
  author = {S. Lanthaler and T. K. Rusch and S. Mishra},
  title = {Neural Oscillators are Universal},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-20.pdf },
  year = {2023}
}

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