Recovery of spatially varying acoustical properties via automated partial differential equation 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. VC 2023 Acoustical Society of America. https://doi.org/10.1121/10.0019592

Automated summary

A data-driven method is proposed to map the spatial variations of physical properties for a material by identifying spatially dependent partial differential equations (PDEs) from observations of dynamical behaviors. This method based on L1-norm minimization does not require any assumed active PDE terms and is capable of identifying spatially dependent PDEs from measurements of phenomena. It has been demonstrated in various experimental settings, including real laser measurements, and is efficient and robust against noise.