Fig. 1. LDNet architecture.

The network ${\mathcal{NN}}_{\mathrm{dyn}}$ receives the input u(t) and the latent state s(t) and returns the time derivative of the latent state, thus defining its dynamics. The network ${\mathcal{NN}}_{\mathrm{rec}}$, instead, is evaluated only when an estimate of the output field y is sought. More precisely, an approximation of y(x, t) is recovered by giving as an input to ${\mathcal{NN}}_{\mathrm{rec}}$ the latent state at time t and the query space coordinate x ∈ Ω. In general, the reconstruction network ${\mathcal{NN}}_{\mathrm{rec}}$ might take as an input u(t) as well (see e.g. Results, Test Case 2); for simplicity, in the figure we represent the special case when ${\mathcal{NN}}_{\mathrm{rec}}$ does not depend on u(t).