  A critical challenge in the data-driven modeling of dynamical systems is
producing methods robust to measurement error, particularly when data is
limited. Many leading methods either rely on denoising prior to learning or on
access to large volumes of data to average over the effect of noise. We propose
a novel paradigm for data-driven modeling that simultaneously learns the
dynamics and estimates the measurement noise at each observation. By
constraining our learning algorithm, our method explicitly accounts for
measurement error in the map between observations, treating both the
measurement error and the dynamics as unknowns to be identified, rather than
assuming idealized noiseless trajectories. We model the unknown vector field
using a deep neural network, imposing a Runge-Kutta integrator structure to
isolate this vector field, even when the data has a non-uniform timestep, thus
constraining and focusing the modeling effort. We demonstrate the ability of
this framework to form predictive models on a variety of canonical test
problems of increasing complexity and show that it is robust to substantial
amounts of measurement error. We also discuss issues with the generalizability
of neural network models for dynamical systems and provide open-source code for
all examples.
