Researchers propose sparse regression for identifying governing partial differential equations for spatiotemporal systems.

In this work we present a data-driven method for the discovery of parametric partial differential equations (PDEs), thus allowing one to disambiguate between the underlying evolution equations and their parametric dependencies. Group sparsity is used to ensure parsimonious representations of observed dynamics in the form of a parametric PDE, while also allowing the coefficients to have arbitrary time series, or spatial dependence. This work builds on previous methods for the identification of constant coefficient PDEs, expanding the field to include a new class of equations which until now have eluded machine learning based identification methods. We show that group sequentially thresholded ridge regression outperforms group LASSO in identifying the fewest terms in the PDE along with their parametric dependency. The method is demonstrated on four canonical models with and without the introduction of noise. Key word. data-driven method, sparse regression, parametric models AMS subject classifications. 37M02, 65P02, 49M02 * Submitted 6/2/2018 Funding: S. Rudy, S. L. Brunton and J. N. Kutz acknowledge support from the Defense Advanced Research Projects Agency (DARPA HR0011-16-C-0016). J. N. Kutz acknowledges support from the Air Force Office of Scientific Research (AFOSR) grant FA9550-15-1-0385. S. L. Brunton acknowledges support from the Air Force Office of Scientific Research (AFOSR FA9550-18-1-0200).

A critical challenge in the data-driven modeling of dynamical systems is producing methods robust to measurement error, particularly when data is limited. Many leading methods either rely on denoising prior to learning or on access to large volumes of data to average over the effect of noise. We propose a novel paradigm for data-driven modeling that simultaneously learns the dynamics and estimates the measurement noise at each observation. By constraining our learning algorithm, our method explicitly accounts for measurement error in the map between observations, treating both the measurement error and the dynamics as unknowns to be identified, rather than assuming idealized noiseless trajectories. We model the unknown vector field using a deep neural network, imposing a Runge-Kutta integrator structure to isolate this vector field, even when the data has a non-uniform timestep, thus constraining and focusing the modeling effort. We demonstrate the ability of this framework to form predictive models on a variety of canonical test problems of increasing complexity and show that it is robust to substantial amounts of measurement error. We also discuss issues with the generalizability of neural network models for dynamical systems and provide open-source code for all examples.