  The study presents a general framework for discovering underlying Partial
Differential Equations (PDEs) using measured spatiotemporal data. The method,
called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides
which physical terms are necessary and which can be removed (because they are
physically negligible in the sense that they do not affect the dynamics too
much) from a pool of candidate functions. The method is built on the recent
development of Sparse Bayesian Learning; which enforces the sparsity in the
to-be-identified PDEs, and therefore can balance the model complexity and
fitting error with theoretical guarantees. Without leveraging prior knowledge
or assumptions in the discovery process, we use an automated approach to
discover ten types of PDEs, including the famous Navier-Stokes and sine-Gordon
equations, from simulation data alone. Moreover, we demonstrate our data-driven
discovery process with the Complex Ginzburg-Landau Equation (CGLE) using data
measured from a traveling-wave convection experiment. Our machine discovery
approach presents solutions that has the potential to inspire, support and
assist physicists for the establishment of physical laws from measured
spatiotemporal data, especially in notorious fields that are often too complex
to allow a straightforward establishment of physical law, such as biophysics,
fluid dynamics, neuroscience or nonlinear optics.
