The wave equation as a differentiable-programmable computer for the design of acoustic metamaterials

Abstract

Designing new metamaterials enables fine-grain control of wave dynamics. Applications of these materials range from wave steering devices for seismic wave manipulation, to “super-focusing” devices for high-resolution ultrasound imaging and surgery. Control of systems governed by partial differential equations is an inherently hard problem. Specifically, control of wave dynamics is a challenging problem, because of the additional physical constraints and intrinsic properties of wave phenomena such as attenuation, reflection, and scattering. In this work, we propose a novel method for transient control of wave equations with Model Predictive Control. The proposed model incorporates the essential physical properties observed in the original dynamics directly into its lower dimensional latent space. We show that our model is capable of using its latent dynamics to predict scalar integral quantities over a variable length timespan. Moreover, it is fully interpretable by the corresponding physical quantities, which allows it to guarantee solution properties. We demonstrate this method by solving important cases of wave equation, which have not been solved before, such as control of transient waves. We make our code publicly available.