Gaussian Process Regression for Data Fulfilling Linear Differential Equations with Localized Sources

Area of Science:

• Applied Mathematics
• Machine Learning
• Scientific Computing

Background:

• Gaussian process regression is a powerful tool for modeling data.
• Existing methods often lack the ability to incorporate known physical constraints like differential equations.
• This limits their applicability in scientific domains where physical laws govern data generation.

Purpose of the Study:

• To develop a specialized Gaussian process regression method tailored for data obeying linear differential equations.
• To enable accurate estimation of system parameters and the location/strength of point sources.
• To ensure generated solutions are physically plausible by design.

Main Methods:

• Restricting Gaussian processes to generate solutions of the homogeneous differential equation using specialized kernels.
• Incorporating point source contributions via a linear model over fundamental solutions.
• Employing maximum likelihood estimation and nonlinear optimization for hyperparameter and source parameter inference.

Main Results:

• Demonstrated accurate modeling of source-free data and parameter estimation for Laplace's and heat/diffusion equations.
• Successfully treated the Helmholtz equation with point sources, applicable to scalar wave data.
• Achieved more reliable regression with less training data compared to generic approaches.

Conclusions:

• The specialized Gaussian process regression provides a physically informed and data-efficient approach for scientific modeling.
• The method generates only physically possible solutions, with estimated hyperparameters representing physical properties.
• Applicable to diverse data from measurements and simulations governed by linear differential equations.