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

TL;DR

This paper introduces a deep learning framework that combines neural networks with Runge-Kutta numerical integration to accurately learn differential equations from noisy, sparsely sampled data, even when variables are measured at different times.

Contribution

The novel integration of neural networks with a Runge-Kutta constraint enables robust learning of dynamical models from noisy, irregularly sampled data, advancing data-driven modeling techniques.

Findings

Effective noise handling in differential equation learning

Can learn from data with asynchronous variable measurements

Demonstrated success on various differential equations

Abstract

Measurement noise is an integral part while collecting data of a physical process. Thus, noise removal is a necessary step to draw conclusions from these data, and it often becomes quite essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and sparsely sampled measurements. In our methodology, the main innovation can be seen in of integration of deep neural networks with a classical 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-steps can be given by following a numerical integration scheme such as the fourth-order Runge-Kutta scheme. The proposed framework to learn a model predicting the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are not available at the same temporal grid. We demonstrate the effectiveness of the proposed method to learning models using data obtained from various differential equations. The proposed approach provides a promising methodology to learn dynamic models, where the first-principle understanding remains opaque.