  To derive the hidden dynamics from observed data is one of the fundamental
but also challenging problems in many different fields. In this study, we
propose a new type of interpretable network called the ordinary differential
equation network (ODENet), in which the numerical integration of explicit
ordinary differential equations (ODEs) are embedded into the machine learning
scheme to build a general framework for revealing the hidden dynamics buried in
massive time-series data efficiently and reliably. ODENet takes full advantage
of both machine learning algorithms and ODE modeling. On one hand, the
embedding of ODEs makes the framework more interpretable benefiting from the
mature theories of ODEs. On the other hand, the schemes of machine learning
enable data handling, paralleling, and optimization to be easily and
efficiently implemented. From classical Lotka-Volterra equations to chaotic
Lorenz equations, the ODENet exhibits its remarkable capability in handling
time-series data even in the presence of large noise. We further apply the
ODENet to real actin aggregation data, which shows an impressive performance as
well. These results demonstrate the superiority of ODENet in dealing with noisy
data, data with either non-equal spacing or large sampling time steps over
other traditional machine learning algorithms.
