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

Uy, Wayne Isaac Tan; Peherstorfer, Benjamin

doi:10.1051/m2an/2021010

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

Journal Article · Wed May 05 00:00:00 EDT 2021 · Mathematical Modelling and Numerical Analysis

DOI:https://doi.org/10.1051/m2an/2021010· OSTI ID:1784724

^[1]; Peherstorfer, Benjamin ^[2]

Courant Inst. of Mathematical Sciences, New York, NY (United States); New York University
Courant Inst. of Mathematical Sciences, New York, NY (United States)

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. Here, 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.

View Accepted Manuscript (DOE)

Research Organization:: New York Univ., NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0019334

OSTI ID:: 1784724

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Journal Issue: 3 Vol. 55; ISSN 0764-583X

Publisher:: EDP SciencesCopyright Statement

Country of Publication:: United States

Language:: English

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