# Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

arXiv: https://arxiv.org/abs/2005.05890 · HTML: https://mlsanity.com/paper/2005.05890/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2020.

## Built on by (most cited first, 6 of 6)

- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories](https://mlsanity.com/paper/2103.01362/) (2021)
- [Active operator inference for learning low-dimensional dynamical-system models from noisy data](https://mlsanity.com/paper/2107.09256/) (2021)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [Rank-Minimizing and Structured Model Inference](https://mlsanity.com/paper/2302.09521/) (2023)

## Abstract (arXiv)

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.