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To derive the hidden dynamics from observed data is one of the fundamentalbut also challenging problems in many different fields. In this study, wepropose a new type of interpretable network called the ordinary differentialequation network (ODENet), in which the numerical integration of explicitordinary differential equations (ODEs) are embedded into the machine learningscheme to build a general framework for revealing the hidden dynamics buried inmassive time-series data efficiently and reliably. ODENet takes full advantageof both machine learning algorithms and ODE modeling. On one hand, theembedding of ODEs makes the framework more interpretable benefiting from themature theories of ODEs. On the other hand, the schemes of machine learningenable data handling, paralleling, and optimization to be easily andefficiently implemented. From classical Lotka-Volterra equations to chaoticLorenz equations, the ODENet exhibits its remarkable capability in handlingtime-series data even in the presence of large noise. We further apply theODENet to real actin aggregation data, which shows an impressive performance aswell. These results demonstrate the superiority of ODENet in dealing with noisydata, data with either non-equal spacing or large sampling time steps overother traditional machine learning algorithms.

Revealing hidden dynamics from time-series data by ODENet