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## Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations

## Abstract

Here, this work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The proposed statistical model comprises the sum of a deterministic regression-function model and a stochastic noise model. The method constructs the regression-function model by applying regression techniques from machine learning (e.g., support vector regression, artificial neural networks) to map features (i.e., error indicators such as sampled elements of the residual) to a prediction of the approximate-solution error. The method constructs the noise model as a mean-zero Gaussian random variable whose variance is computed as the sample variance of the approximate-solution error on a test set; this variance can be interpreted as the epistemic uncertainty introduced by the approximate solution. This work considers a wide range of feature-engineering methods, data-set-construction techniques, and regression techniques that aim to ensure that (1) the features are cheaply computable, (2) the noise model exhibits low variance (i.e., low epistemic uncertainty introduced), and (3) the regression model generalizes to independent test data. Finally, numerical experiments performed on several computational-mechanics problems andmore »

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

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

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

- OSTI Identifier:
- 1498489

- Alternate Identifier(s):
- OSTI ID: 1498763

- Report Number(s):
- SAND-2018-13294J; SAND-2018-8670J

Journal ID: ISSN 0045-7825; 670277

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 348; Journal Issue: C; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Error modeling; Supervised machine learning; High-dimensional regression; Parameterized nonlinear equations; Model reduction; ROMES method

### Citation Formats

```
Freno, Brian Andrew, and Carlberg, Kevin Thomas. Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations. United States: N. p., 2019.
Web. doi:10.1016/j.cma.2019.01.024.
```

```
Freno, Brian Andrew, & Carlberg, Kevin Thomas. Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations. United States. doi:10.1016/j.cma.2019.01.024.
```

```
Freno, Brian Andrew, and Carlberg, Kevin Thomas. Tue .
"Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations". United States. doi:10.1016/j.cma.2019.01.024.
```

```
@article{osti_1498489,
```

title = {Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations},

author = {Freno, Brian Andrew and Carlberg, Kevin Thomas},

abstractNote = {Here, this work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The proposed statistical model comprises the sum of a deterministic regression-function model and a stochastic noise model. The method constructs the regression-function model by applying regression techniques from machine learning (e.g., support vector regression, artificial neural networks) to map features (i.e., error indicators such as sampled elements of the residual) to a prediction of the approximate-solution error. The method constructs the noise model as a mean-zero Gaussian random variable whose variance is computed as the sample variance of the approximate-solution error on a test set; this variance can be interpreted as the epistemic uncertainty introduced by the approximate solution. This work considers a wide range of feature-engineering methods, data-set-construction techniques, and regression techniques that aim to ensure that (1) the features are cheaply computable, (2) the noise model exhibits low variance (i.e., low epistemic uncertainty introduced), and (3) the regression model generalizes to independent test data. Finally, numerical experiments performed on several computational-mechanics problems and types of approximate solutions demonstrate the ability of the method to generate statistical models of the error that satisfy these criteria and significantly outperform more commonly adopted approaches for error modeling.},

doi = {10.1016/j.cma.2019.01.024},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = C,

volume = 348,

place = {United States},

year = {2019},

month = {2}

}