Deep learning of dynamics and signal-noise decomposition with time-stepping constraints

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. 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 neural network, imposing a Runge-Kutta integrator structure to isolate this vector field, even when the data has a non-uniform time-step, thus constraining and focusing the modeling effort. We demonstrate the ability of this framework to form predictive models on a variety of test problems including low dimensional flow around a cylinder and discuss some challenges with using neural networks to interpolate governing equations.