  This paper investigates how models of spatiotemporal dynamics in the form of
nonlinear partial differential equations can be identified directly from noisy
data using a combination of sparse regression and weak formulation. Using the
4th-order Kuramoto-Sivashinsky equation for illustration, we show how this
approach can be optimized in the limits of low and high noise, achieving
accuracy that is orders of magnitude better than what existing techniques
allow. In particular, we derive the scaling relation between the accuracy of
the model, the parameters of the weak formulation, and the properties of the
data, such as its spatial and temporal resolution and the level of noise.
