Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers

TL;DR

The results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.

Abstract

Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.

Figures (9)

• Figure 1: A physics-informed Gaussian process framework for the solution of linear PDEs.
• Figure 2: Physics-informed Gaussian process model of the stationary temperature distribution in an idealized hexa-core CPU die under sustained computational load.
• Figure 3: Prior model for the stationary temperature distribution of a CPU die under load.
• Figure 4: We integrate mechanistic knowledge about the system by conditioning on PDE observations $-\kappa \IfNoValueTF{-NoValue-}{ \Delta {\mathrm{u}} \IfNoValueF{{\bm{X}}_{\text{PDE}}}{\left( {\bm{X}}_{\text{PDE}} \right)} }{ \left. \Delta {\mathrm{u}} \right|_{-NoValue-\IfNoValueF{{\bm{X}}_{\text{PDE}}}{= {\bm{X}}_{\text{PDE}}}} } - \dot{q}_V({\bm{X}}_{\text{PDE}}) = {\bm{0}}$ at the collocation points ${\bm{X}}_{\text{PDE}}$, resulting in the conditional process $\IfNoValueTF{-NoValue-}{ { {\mathrm{u}} \nonscript\:\vert \nonscript\: \mathopen{} \text{PDE} } }{ \@condrv[-NoValue-]{{\mathrm{u}} \nonscript\:\vert \nonscript\: \mathopen{} \text{PDE}} }$. The large remaining uncertainty in \ref{['subfig:cpu-stationary-1d-cond-pde-belief-solution']} illustrates that the PDE by itself does not identify a unique solution.
• Figure 5: Neumann boundary conditions encoding mechanistic knowledge about the heat flux across the boundary of the CPU and a sparse set of limited-precision measurements of the temperature distribution made by digital thermal sensors (DTS) located at the points ${\bm{X}}_{\text{DTS}}$ further constrain the solution of the PDE. The remaining uncertainty is due to measurement noise and discretization error.

Theorems & Definitions (34)

• definition 1
• example 1: Thermal Conduction and the Heat Equation
• example 2: Stationary Heat Equation
• example 3: continues=ex:thermal-conduction-heat-equation
• example 4: Symmetric Collocation
• example 5: Weak Formulations
• example 6
• example 7: A 1D Finite Element Method
• definition 2: MWR Information Operator
• proposition 3