Inspired by recent developments in data-driven methods for partial differential equation (PDE) estimation, we use sparse modeling techniques to automatically estimate PDEs from data. A dictionary consisting of hypothetical PDE terms is constructed using numerical differentiation. Given data, PDE terms are selected assuming a parsimonious representation, which is enforced using a sparsity constraint. Unlike previous PDE identification schemes, we make no assumptions about which PDE terms are responsible for a given field. The approach is demonstrated on synthetic and real video data, with physical phenomena governed by wave, Burgers, and Helmholtz equations. Codes are available at https://github.com/NoiseLab-RLiu/Automate-PDE-identification.

Observable dynamics, such as waves propagating on a surface, are generally governed by partial differential equations (PDEs), which are determined by the physical properties of the propagation media. The spatial variations of these properties lead to spatially dependent PDEs. It is useful in many fields to recover the variations from the observations of dynamical behaviors on the material. A method is proposed to form a map of the physical properties' spatial variations for a material via data-driven spatially dependent PDE identification and applied to recover acoustical properties (viscosity, attenuation, and phase speeds) for propagating waves. The proposed data-driven PDE identification scheme is based on ℓ1-norm minimization. It does not require any PDE term that is assumed active from the prior knowledge and is the first approach that is capable of identifying spatially dependent PDEs from measurements of phenomena. In addition, the method is efficient as a result of its non-iterative nature and can be robust against noise if used with an integration transformation technique. It is demonstrated in multiple experimental settings, including real laser measurements of a vibrating aluminum plate. Codes and data are available online at https://tinyurl.com/4wza8vxs.

We use a machine learning-based tomography method to obtain high-resolution subsurface geophysical structure in Long Beach, CA, from seismic noise recorded on a "large-N" array with 5204 geophones (~13.5 million travel times). This method, called locally sparse travel time tomography (LST) uses unsupervised machine learning to exploit the dense sampling obtained by ambient noise processing on large arrays. Dense sampling permits the LST method to learn directly from the data a dictionary of local, or small-scale, geophysical features. The features are the small scale patterns of Earth structure most relevant to the given tomographic imaging scenario. Using LST, we obtain a high-resolution 1 Hz Rayleigh wave phase speed map of Long Beach. Among the geophysical features shown in the map, the important Silverado aquifer is well isolated relative to previous surface wave tomography studies. Our results show promise for LST in obtaining detailed geophysical structure in travel time tomography studies.