  Continuous-time (CT) modeling has proven to provide improved sample
efficiency and interpretability in learning the dynamical behavior of physical
systems compared to discrete-time (DT) models. However, even with numerous
recent developments, the CT nonlinear state-space (NL-SS) model identification
problem remains to be solved in full, considering common experimental aspects
such as the presence of external inputs, measurement noise, latent states, and
general robustness. This paper presents a novel estimation method that
addresses all these aspects and that can obtain state-of-the-art results on
multiple benchmarks with compact fully connected neural networks capturing the
CT dynamics. The proposed estimation method called the subspace encoder
approach (SUBNET) ascertains these results by efficiently approximating the
complete simulation loss by evaluating short simulations on subsections of the
data, by using an encoder function to estimate the initial state for each
subsection and a novel state-derivative normalization to ensure stability and
good numerical conditioning of the training process. We prove that the use of
subsections increases cost function smoothness together with the necessary
requirements for the existence of the encoder function and we show that the
proposed state-derivative normalization is essential for reliable estimation of
CT NL-SS models.
