Overview
This research addresses the closure problem inherent in Reduced-Order Models (ROMs), which often exhibit compromised predictive accuracy due to truncation errors and inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) scales on ROM (resolved) scales constitutes the closure problem. The study formulates ROM closure modeling as a multi-fidelity (MF) learning problem and introduces an uncertainty-aware MF framework. This framework, based on conditional normalizing flows, aims to enhance ROM predictive accuracy.
Research Context
Reduced-Order Models serve as efficient surrogates for complex multiscale systems. However, their utility is constrained by limitations in predictive accuracy. These limitations stem from two primary factors: truncation errors and insufficient representation of interactions between resolved and unresolved scales. The specific challenge of accounting for the influence of truncated, or unresolved, scales on the resolved scales of a ROM is termed the closure problem.
Approach
The proposed methodology frames the ROM closure modeling challenge as a multi-fidelity learning problem. It leverages conditional normalizing flows to establish an uncertainty-aware MF framework. This framework is designed to learn a probabilistic mapping between low-fidelity (LF) ROM coefficients and high-fidelity (HF) coefficients. The objective of this mapping is to improve predictive fidelity while simultaneously quantifying the uncertainty associated with the learned closure.
Two distinct correction strategies were investigated within this framework:
- Direct Learning: In this strategy, high-fidelity coefficients are predicted directly from the low-fidelity inputs.
- Residual Learning: This approach focuses on learning the discrepancy, or residual, between the low-fidelity and high-fidelity coefficients.
The framework's efficacy was demonstrated using two specific problems governed by the two-dimensional Navier-Stokes equations: a double shear layer problem and vortex merging problems.
Findings
The application of the proposed framework yielded several key observations:
- Both the direct learning and residual learning correction strategies improved the accuracy of the ROMs when compared to uncorrected ROMs.
- Residual learning consistently achieved superior performance compared to direct learning in enhancing ROM accuracy.
- The two deep generative model-based strategies, direct learning and residual learning, provide uncertainty quantification for the corrected ROM coefficients. This quantification is identified as critical for assessing prediction confidence.
- The quantification of uncertainty supports the reliable use of ROMs in practical applications.
Why This Matters
The developed framework enhances the predictive fidelity of Reduced-Order Models while quantifying associated uncertainties. This capability for uncertainty quantification is crucial for assessing prediction confidence, thereby supporting the reliable application of ROMs in practical contexts.
Potential Applications
The ability to quantify uncertainty in corrected ROM coefficients is presented as critical for assessing prediction confidence and supporting the reliable use of ROMs in practical applications.