Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint

Measurement noise is an integral part while collecting data of physical processes. Thus, noise removal is necessary to draw conclusions from these data and is essential to construct dynamic models using these data. This work discusses a methodology for learning dynamic models using noisy measurements and simultaneously obtaining denoised data. In our methodology, the main innovation can be seen in integrating deep neural networks with a numerical integration method. Precisely, we aim at learning a neural network that implicitly represents the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by enforcing the constraint that the data at the next time-step can be obtained by following a numerical integration scheme. The proposed framework to identify a model predicting the vector field is effective under noisy measurements and provides denoised data. We demonstrate the effectiveness of the proposed method to learn models using a differential equation and present a comparison with the neural ODE approach.