Structured neural networks motivated by dynamical systems

Davide Murari

Math Meets Industry - 02/06/2022



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

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