PDE-constrained Gaussian process surrogate modeling with uncertain data locations
This work addresses surrogate modeling for PDEs with uncertain data locations, which is incremental as it extends Gaussian process methods to handle input variability in computational science and engineering.
The authors tackled the problem of surrogate modeling for partial differential equations with uncertain input data by proposing a Bayesian Gaussian process approach that integrates uncertain inputs via marginalization, achieving a substantial reduction in predictive uncertainties in numerical examples like the heat and Allen-Cahn equations.
Gaussian process regression is widely applied in computational science and engineering for surrogate modeling owning to its kernel-based and probabilistic nature. In this work, we propose a Bayesian approach that integrates the variability of input data into the Gaussian process regression for function and partial differential equation approximation. Leveraging two types of observables -- noise-corrupted outputs with certain inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution of uncertain inputs is estimated via Bayesian inference. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The setting of two types of data aligned with common scenarios of constructing surrogate models for the solutions of partial differential equations, where the data of boundary conditions and initial conditions are typically known while the data of solution may involve uncertainties due to the measurement or stochasticity. The effectiveness of the proposed method is demonstrated through several numerical examples including multiple one-dimensional functions, the heat equation and Allen-Cahn equation. A consistently good performance of generalization is observed, and a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.