Conditional Normalizing Flows Boost Reduced-Order Model Accuracy with Uncertainty Quantification

Reduced-order models (ROMs) are essential for simulating complex multiscale systems at a fraction of the computational cost of full-order models, but they suffer from truncation errors when unresolved scales are omitted. This 'closure problem' limits predictive accuracy. In a new arXiv paper (2606.09857), Jice Zeng and co-authors treat ROM closure as a multi-fidelity learning problem and use conditional normalizing flows—a type of deep generative model—to map low-fidelity (LF) ROM coefficients to high-fidelity (HF) coefficients. The approach provides a probabilistic correction that inherently quantifies uncertainty, a critical feature for reliable predictions. The study compares two correction strategies: direct learning (predicting HF coefficients from LF inputs) and residual learning (learning the discrepancy between LF and HF coefficients).

Using a vortex merging problem governed by the 2D Navier-Stokes equations as a testbed, the authors show that both strategies improve ROM accuracy over uncorrected baselines, but residual learning consistently delivers superior performance. By outputting a distribution rather than a point estimate, the conditional normalizing flow framework allows practitioners to assess prediction confidence, enabling safer deployment of ROMs in engineering and scientific applications. This work demonstrates a promising avenue for fusing data from different fidelity levels with deep generative modeling to enhance surrogate models while maintaining uncertainty awareness.