Deep limits of residual neural networks

Deep limits of residual neural networks

Thorpe, Matthew (The University of Manchester; The Alan Turing Institute)
van Gennip, Y. (TU Delft Mathematical Physics)

2023

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The contribution of this paper is to show that, for the residual neural network model, the deep layer limit coincides with a parameter estimation problem for a nonlinear ordinary differential equation. In particular, whilst it is known that the residual neural network model is a discretisation of an ordinary differential equation, we show convergence in a variational sense. This implies that optimal parameters converge in the deep layer limit. This is a stronger statement than saying for a fixed parameter the residual neural network model converges (the latter does not in general imply the former). Our variational analysis provides a discrete-to-continuum Γ -convergence result for the objective function of the residual neural network training step to a variational problem constrained by a system of ordinary differential equations; this rigorously connects the discrete setting to a continuum problem.

Deep layer limits
Deep neural networks
Gamma-convergence
Ordinary differential equations
Regularity
Variational convergence

http://resolver.tudelft.nl/uuid:d6fa0e23-1c2b-48f2-a676-2d843b109342

https://doi.org/10.1007/s40687-022-00370-y

2522-0144

Research in Mathematical Sciences, 10 (1)

Institutional Repository

journal article

© 2023 Matthew Thorpe, Y. van Gennip