Structured neural networks motivated by dynamical systems
Davide Murari
Math Meets Industry - 02/06/2022
\(\texttt{davide.murari@ntnu.no}\)
Motivation
What are neural networks
\( \mathcal{NN}(x) = f_{\theta_k}\circ ... \circ f_{\theta_1}(x)\)
Dynamical systems interpretation
\( \dot{x}(t) = f(t,x(t),\theta(t)) \)
\mathcal{NN}(x) = \Psi_{f_k}^{h_k}\circ ...\circ \Psi_{f_1}^{h_1}(x)
Layers with a prescribed property \(\mathcal{P}\)
Choose \(f\)
with trajectories that satisfy \(\mathcal{P}\)
Choose \(\Psi^{h_i}_{f_i}\)
that preserves \(\mathcal{P}\)
\(i-\)th layer \(x\mapsto \Psi_{f_i}^{h_i}(x)\)
satisfies \(\mathcal{P}\)
Example : Mass preserving networks
Thank you for the attention