Neural ordinary differential equations with irregular and noisy data

Measurement noise is an integral part of collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and irregularly sampled measurements. In our methodology, the main innovation can be seen in the integration of deep neural networks with the neural ordinary differential equations (ODEs) approach. Precisely, we aim at learning a neural network that provides (approximately) an implicit representation of the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by constraints using neural ODEs. The proposed framework to learn a model describing the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are unavailable at the same temporal grid. Moreover, a particular structure, e.g. second order with respect to time, can easily be incorporated. We demonstrate the effectiveness of the proposed method for learning models using data obtained from various differential equations and present a comparison with the neural ODE method that does not make any special treatment to noise. Additionally, we discuss an ensemble approach to improve the performance of the proposed approach further.

Automated summary

Measurement noise is a crucial factor in data collection for physical processes. This study introduces a methodology that combines deep neural networks with neural ordinary differential equations to learn differential equations from noisy and irregularly sampled measurements. The proposed framework effectively models vector fields under noisy measurements and can handle scenarios with unavailable dependent variables on the same temporal grid. The method is demonstrated to be effective for learning models from data obtained from various differential equations and performs better than the neural ordinary differential equation method without special treatment to noise. An ensemble approach is also discussed to further improve the performance of the proposed method.