Deep Gaussian Processes and Multivariate Bayesian Spline-Based Methods for Emulating Nonlinear Cardiovascular Systems

This study compares Gaussian processes (GPs), two-layer deep Gaussian processes (DGPs), and Bayesian multivariate tensor-product splines as emulators.The methods are first assessed on two toy functions across sample sizes 𝑛 = {100,250,500,1000}.They are then applied to two biomechanical simulators of increasing complexity: a 1D haemodynamic model, emulated through a scalar objective measuring mismatch to a reference simulation, and a 3D finite-element model of a stenosed artery with axially symmetric plaque, emulated through nodal displacement from systole to diastole.For the toy functions, DGPs gave the strongest predictive accuracy in smaller samples.However, when larger samples required the Vecchia sparse covariance approximation, DGP accuracy fell below the other models.This loss of performance appears to be linked to poor mixing of latent hyperparameters.DGPs were also substantially more computationally expensive, with computation times around ten times higher than the comparison methods and lower MCMC effective sample sizes.Splines were the most accurate predictors at large samples and were the most computationally efficient in general.In the 1D haemodynamic case, GPs and DGPs performed comparably, while the spline emulator was less accurate.In the 3D finite-element case, where the target surface showed non-stationary smoothness across the input domain, the DGP achieved the strongest overall predictive accuracy.Post-hoc analysis suggests that GP and spline models were less accurate where the response surface had non-uniform gradients, which could not be represented by a single smoothness regime.However, the DGP sometimes introduced spurious curvature in flatter regions of the target surface.Overall, DGPs were most reliable for emulating non-stationarity in the response surfaces, particularly in smaller samples.However, their predictive advantage was not retained for larger samples requiring the Vecchia approximation.Splines demonstrated the most attractive balance of prediction accuracy and efficiency for larger samples.