Gradient-enhanced deep Gaussian processes for multifidelity modelling

TL;DR

This work extends multifidelity Gaussian process modelling by integrating gradient information into gradient-enhanced deep GPs, enabling nonlinear, input-dependent fidelity relationships to be captured. The authors develop gradient kernels and an augmented variational framework with inducing gradients to perform scalable inference via sparse variational methods, including an alternative predictive log-likelihood objective. They demonstrate on a Branin-type analytical benchmark and a hypersonic PDE problem that gradient-enhanced deep GPs outperform gradient-enhanced LMC and non-gradient models, with PLL often offering richer uncertainty quantification. The method, implemented in gpytorch with Adam optimization, shows promise for efficient surrogate modelling in PDE solvers and aerospace design problems, suggesting potential extensions to larger datasets and gradient-based optimization tasks.

Abstract

Multifidelity models integrate data from multiple sources to produce a single approximator for the underlying process. Dense low-fidelity samples are used to reduce interpolation error, while sparse high-fidelity samples are used to compensate for bias or noise in the low-fidelity samples. Deep Gaussian processes (GPs) are attractive for multifidelity modelling as they are non-parametric, robust to overfitting, perform well for small datasets, and, critically, can capture nonlinear and input-dependent relationships between data of different fidelities. Many datasets naturally contain gradient data, especially when they are generated by computational models that are compatible with automatic differentiation or have adjoint solutions. Principally, this work extends deep GPs to incorporate gradient data. We demonstrate this method on an analytical test problem and a realistic partial differential equation problem, where we predict the aerodynamic coefficients of a hypersonic flight vehicle over a range of flight conditions and geometries. In both examples, the gradient-enhanced deep GP outperforms a gradient-enhanced linear GP model and their non-gradient-enhanced counterparts.

Figures (9)

• Figure 1: Multifidelity Branin function
• Figure 2: Prediction errors on multifidelity Branin function, medium sampling
• Figure 3: Performance of gradient-enhanced GP models on the multifidelity Branin test problem, 'medium' sampling density
• Figure 4: Hypersonic waverider geometry
• Figure 5: Body curvature (exaggerated)