Noise-robust multi-fidelity surrogate modelling for parametric partial differential equations

Abstract: We address the challenge of constructing noise-robust surrogate models for quantities of interest (QoIs) arising from parametric partial differential equations (PDEs), using multi-fidelity collocation techniques; specifically, the Multi-Index Stochastic Collocation (MISC). In practical scenarios, the PDE evaluations used to build a response surface are often corrupted by numerical noise, especially for the low-fidelity models. This noise, which may originate from loose solver tolerances, coarse discretisations, or transient effects, can lead to overfitting in MISC, degrading surrogate quality through nonphysical oscillations and loss of convergence, thereby limiting its utility in downstream tasks like uncertainty quantification, optimisation, and control. To correct this behaviour, we propose an improved version of MISC that can automatically detect the presence of solver noise during the surrogate model construction and then ignore the exhausted fidelities. Our approach monitors the spectral decay of the surrogate at each iteration, identifying stagnation in the coefficient spectrum that signals the onset of noise. Once detected, the algorithm selectively halts the use of noisy fidelities, focusing computational resources on those fidelities that still provide meaningful information. The effectiveness of this approach is numerically validated on two challenging test cases: a parabolic advection--diffusion PDE with uncertain coefficients, and a parametric turbulent incompressible Navier--Stokes problem. The results showcase the accuracy and robustness of the resulting multi-fidelity surrogate and its capability to extract relevant information, even from under-resolved meshes not suitable for reliable single-fidelity computations.