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Measurement noise is an integral part while collecting data of a physicalprocess. Thus, noise removal is a necessary step to draw conclusions from thesedata, and it often becomes quite essential to construct dynamical models usingthese data. We discuss a methodology to learn differential equation(s) usingnoisy and sparsely sampled measurements. In our methodology, the maininnovation can be seen in of integration of deep neural networks with aclassical numerical integration method. Precisely, we aim at learning a neuralnetwork that implicitly represents the data and an additional neural networkthat models the vector fields of the dependent variables. We combine these twonetworks by enforcing the constraint that the data at the next time-steps canbe given by following a numerical integration scheme such as the fourth-orderRunge-Kutta scheme. The proposed framework to learn a model predicting thevector field is highly effective under noisy measurements. The approach canhandle scenarios where dependent variables are not available at the sametemporal grid. We demonstrate the effectiveness of the proposed method tolearning models using data obtained from various differential equations. Theproposed approach provides a promising methodology to learn dynamic models,where the first-principle understanding remains opaque.

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