Comparison of Methods for Learning Differential Equations from Data

Some results from the DEEB (Differential Equation Estimation Benchmark) are presented. In DEEB, we compare different machine learning approaches and statistical methods for estimating nonlinear dynamics from data. Such methods constitute an important building block for purely data-driven earth system models as well as hybrid models which combine physical knowledge with past observations.

Specifically, we examine approaches for solving the following problem: Given time-state-observations of a deterministic ordinary differential equation (ODE) with measurement noise in the state, predict the future evolution of the system. Of particular interest are systems with chaotic behavior - like Lorenz 63 - and nonparametric settings, in which the functional form of the ODE is completely unknown (in particular, not restricted to a polynomial of low order). To create a fair comparison of methods, a benchmark database was created which includes datasets of simulated observations from different dynamical systems with different complexity and varying noise levels. The list of methods we compare includes: echo state networks, Gaussian processes, Neural ODEs, SINDy, thin plate splines, and more.

Although some methods consistently perform better than others throughout different datasets, there seems to be no silver bullet.