Functional Regression for State Prediction Using Linear PDE Models and Observations

Partial differential equations (PDEs) are commonly used to model a wide variety of physical phenomena. A PDE model of a physical problem is typically described by conservation laws, constitutive laws, material properties, boundary conditions, boundary data, and geometry. In most practical applications, however, the PDE model is only an approximation to the real physical problem due to both (i) the deliberate mathematical simplification of the model to keep it tractable, and (ii) the inherent uncertainty of the physical parameters. In such cases, the PDE model may not produce a good prediction of the true state of the underlying physical problem. In this paper, we introduce a Gaussian functional regression method that incorporates observations into a deterministic linear PDE model to improve its prediction of the true state. Our method is devised as follows. First, we augment the PDE model with a random Gaussian functional which serves to represent various sources of uncertainty in the model. We next derive a linear regression model for the Gaussian functional by utilizing observations and adjoint states. We then apply the principle of Gaussian process to determine the posterior distribution of the Gaussian functional. This allows us to compute the posterior distribution for our estimate of the true state and provide a statistical measure of the prediction error. Furthermore, we consider the problem of experimental design in this setting, wherein we develop an algorithm for designing experiments to efficiently reduce the variance of our state estimate. We provide several examples from heat conduction, the convection-diffusion equation, and the reduced wave equation, all of which that demonstrate the performance of the proposed methodology. ∗MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA 02139, USA, email: [email protected] †MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA 02139, USA, email: [email protected] ‡MIT Sloan School of Management, 77 Massachusetts Ave., Cambridge, MA 02139, USA, email: [email protected] §MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA 02139, USA, email: [email protected]