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

2025-07-04

AI Summary

Low-fidelity models for parametric partial differential equations (PDEs) are often corrupted by numerical noise—e.g., from relaxed solver tolerances or coarse discretizations—causing Multi-Index Stochastic Collocation (MISC) to overfit, exhibit unphysical oscillations, and fail to converge. Method: We propose a noise-robust multi-fidelity surrogate modeling framework. Its core innovations include: (i) a spectral decay–based automatic noise detection mechanism that dynamically masks fidelity levels with low signal-to-noise ratios; and (ii) integration of adaptive multi-fidelity sampling with a modified MISC formulation to ensure robust collocation. Results: Evaluated on convection-diffusion and turbulent Navier–Stokes problems, the method effectively suppresses spurious oscillations, restores high-order convergence, and reliably extracts response information even on coarse grids. This enhances the accuracy and stability of downstream uncertainty quantification and optimization tasks.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Construct noise-robust surrogates for parametric PDEs
• Mitigate overfitting from numerical noise in multi-fidelity models
• Improve surrogate quality for uncertainty quantification and optimization

Innovation

• Noise-robust multi-fidelity surrogate modeling
• Automatic detection of solver noise
• Selective halting of noisy fidelities