Gaussian Process Regression for Inverse Problems in Linear PDEs
This addresses inverse problems in physics and engineering with a novel computational approach, though it appears incremental as it builds on existing Gaussian process and algebraic methods.
The paper tackles inverse problems in linear PDEs by introducing a computationally efficient algorithm using Gaussian process regression with priors from commutative algebra, achieving high accuracy in applications like wave speed identification.
This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency.