Gaussian process regression as a surrogate model for the computation of dispersion relations

The ability to design materials for wave propagation behaviors has high potential for impact in medical imaging, telecommunications, and signal processing. The dispersion relation is the key mathematical object that describes linear elastic wave propagation behavior in a material. To date, the design of wave propagation behavior in materials has been limited to one or two design objectives, for example, designing for one or two bandgaps or wavespeeds. This is a result of the complicated relationship between a material and its dispersion relation, and the hefty computational cost of traditional dispersion computations. Decreasing the amount of time required to perform dispersion computations will streamline the design process of wave devices, allowing for more thorough design optimization workflows, more highly resolved design features, and multi -functional materials. In this work, we demonstrate a specialized Gaussian Process Regressor as a surrogate model for the computation of dispersion relations to alleviate the immense computational cost of the traditional model. By measuring covariance information from relatively small training sets (10s to 1000s of dispersion relations), our surrogate model efficiently infers the full dispersion relation after solving for the dispersion relation at only a sparse set of wavevector points. Further, we provide a mathematical framework to use this model for material design through gradient -based optimization methods.