# Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems

## Abstract

A machine learning–based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high-dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed “error indicators” (ie, features) produced by the surrogate model at a given time instance to a prediction of the surrogate-model error in a quantity of interest (QoI). This eliminates the need for the user to hand-select a small number of informative features. The methodology requires a training set of parameter instances at which the time-dependent surrogate-model error is computed by simulating both the high-fidelity and surrogate models. Using these training data, the method first determines regression-model locality (via classification or clustering) and subsequently constructs a “local” regression model to predict the time-instantaneous error within each identified region of feature space. We consider 2 uses for the resulting error model: (1) as a correction to the surrogate-model QoI prediction at each time instance and (2) as a way to statistically model arbitrary functions of the time-dependent surrogate-model error (eg, time-integrated errors). We then apply the proposed framework to model errors in reduced-order models of nonlinear oil-water subsurface flow simulations, with time-varyingmore »

- Authors:

- Stanford Univ., CA (United States). Dept. of Energy Resources Engineering
- Sandia National Lab. (SNL-CA), Livermore, CA (United States). Extreme-Scae Data Science and Analytics Dept.

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1399882

- Report Number(s):
- SAND-2016-12535J

Journal ID: ISSN 0029-5981; 649856; TRN: US1702976

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal for Numerical Methods in Engineering

- Additional Journal Information:
- Journal Volume: 112; Journal Issue: 12; Journal ID: ISSN 0029-5981

- Publisher:
- Wiley

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; error modeling; machine learning; nonlinar dynamical system; POD-TPWL; surrogate model

### Citation Formats

```
Trehan, Sumeet, Carlberg, Kevin T., and Durlofsky, Louis J.
```*Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems*. United States: N. p., 2017.
Web. doi:10.1002/nme.5583.

```
Trehan, Sumeet, Carlberg, Kevin T., & Durlofsky, Louis J.
```*Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems*. United States. doi:10.1002/nme.5583.

```
Trehan, Sumeet, Carlberg, Kevin T., and Durlofsky, Louis J. Fri .
"Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems". United States.
doi:10.1002/nme.5583. https://www.osti.gov/servlets/purl/1399882.
```

```
@article{osti_1399882,
```

title = {Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems},

author = {Trehan, Sumeet and Carlberg, Kevin T. and Durlofsky, Louis J.},

abstractNote = {A machine learning–based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high-dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed “error indicators” (ie, features) produced by the surrogate model at a given time instance to a prediction of the surrogate-model error in a quantity of interest (QoI). This eliminates the need for the user to hand-select a small number of informative features. The methodology requires a training set of parameter instances at which the time-dependent surrogate-model error is computed by simulating both the high-fidelity and surrogate models. Using these training data, the method first determines regression-model locality (via classification or clustering) and subsequently constructs a “local” regression model to predict the time-instantaneous error within each identified region of feature space. We consider 2 uses for the resulting error model: (1) as a correction to the surrogate-model QoI prediction at each time instance and (2) as a way to statistically model arbitrary functions of the time-dependent surrogate-model error (eg, time-integrated errors). We then apply the proposed framework to model errors in reduced-order models of nonlinear oil-water subsurface flow simulations, with time-varying well-control (bottom-hole pressure) parameters. The reduced-order models used in this work entail application of trajectory piecewise linearization in conjunction with proper orthogonal decomposition. Moreover, when the first use of the method is considered, numerical experiments demonstrate consistent improvement in accuracy in the time-instantaneous QoI prediction relative to the original surrogate model, across a large number of test cases. When the second use is considered, results show that the proposed method provides accurate statistical predictions of the time- and well-averaged errors.},

doi = {10.1002/nme.5583},

journal = {International Journal for Numerical Methods in Engineering},

number = 12,

volume = 112,

place = {United States},

year = {Fri Jul 14 00:00:00 EDT 2017},

month = {Fri Jul 14 00:00:00 EDT 2017}

}

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