  Sparse regression has recently emerged as an attractive approach for
discovering models of spatiotemporally complex dynamics directly from data. In
many instances, such models are in the form of nonlinear partial differential
equations (PDEs); hence sparse regression typically requires evaluation of
various partial derivatives. However, accurate evaluation of derivatives,
especially of high order, is infeasible when the data are noisy, which has a
dramatic negative effect on the result of regression. We present a novel and
rather general approach that addresses this difficulty by using a weak
formulation of the problem. For instance, it allows accurate reconstruction of
PDEs involving high-order derivatives, such as the Kuramoto-Sivashinsky
equation, from data with a considerable amount of noise. The flexibility of our
approach also allows reconstruction of PDE models that involve latent variables
which cannot be measured directly with acceptable accuracy. This is illustrated
by reconstructing a model for a weakly turbulent flow in a thin fluid layer,
where neither the forcing nor the pressure field is known.
