Filtered Neural Galerkin model reduction schemes for efficient propagation of initial condition uncertainties in digital twins

TL;DR

This work tackles the high cost of propagating initial-condition uncertainties in digital twins by introducing filtered Neural Galerkin (FNG) schemes. By propagating the mean and covariance of the reduced-model weight distribution through a Gaussian-moment closure of pre-trained Neural Galerkin dynamics, the method avoids large ensembles while preserving uncertainty information, akin to Gaussian filtering and extended Kalman filtering. The approach combines flow-based weight distributions, moment closures, and Bayesian pre-training of CoLoRA networks to yield an online, low-dimensional, closed-form uncertainty propagation with costs scaling with the online-dimension $p$. Numerical experiments on Burgers and Vlasov problems demonstrate comparable accuracy to ensemble methods but with more than an order of magnitude speedup, highlighting practical impact for control and data-assimilation loops in digital twins.

Abstract

Uncertainty quantification in digital twins is critical to enable reliable and credible predictions beyond available data. A key challenge is that ensemble-based approaches can become prohibitively expensive when embedded in control and data assimilation loops in digital twins, even when reduced models are used. We introduce a reduced modeling approach that advances in time the mean and covariance of the reduced solution distribution induced by the initial condition uncertainties, which eliminates the need to maintain and propagate a costly ensemble of reduced solutions. The mean and covariance dynamics are obtained as a moment closure from Neural Galerkin schemes on pre-trained neural networks, which can be interpreted as filtered Neural Galerkin dynamics analogous to Gaussian filtering and the extended Kalman filter. Numerical experiments demonstrate that filtered Neural Galerkin schemes achieve more than one order of magnitude speedup compared to ensemble-based uncertainty propagation.

Figures (7)

• Figure 1: Burgers: A solution of the filtered Neural Galerkin (NG) model provides comparable quantile intervals (QI) as an ensemble of 100 Neural Galerkin solutions for different realizations of the initial condition in this example.
• Figure 2: Burgers: The filtered Neural Galerkin approach approximates well the mean and quantile interval (QI) that is generated by an ensemble of 100 Neural Galerkin solutions.
• Figure 3: Burgers: At least 50 ensemble members are needed to achieve a comparable accuracy as the quantile interval (QI) predicted by filtered Neural Galerkin schemes. Time is $t =0.5$.
• Figure 4: Burgers: Shows that the width of the quantile intervals computed with filtered Neural Galerkin and an ensemble of Neural Galerkin solutions starts to agree when at least 50 ensemble members are used.
• Figure 5: Burgers: Filtered Neural Galerkin generates quantile intervals with more than one order of magnitude lower runtime compared to generating quantile intervals with 100 ensembles members (EM) of Neural Galerkin solutions.