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

Freno, Brian A.; Carlberg, Kevin

doi:10.1016/j.cma.2019.01.024

Cited by 32 publications

(38 citation statements)

References 76 publications

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“…wheres n : D → R. As in Ref. [15], we now discuss three types of surrogate models that are often employed to generate the approximate solutionsx n (µ), n = 0, . .…”

Section: Approximate Solutionsmentioning

confidence: 99%

“…2. The error bounds exhibit dependence on non-local quantities, i.e., the error at the nth time instance depends on the error at the previous k n time instances through recursion via the rightmost term in inequalities (14) and (15). This comprises a fundamental difference from error bounds for static problems.…”

Section: A Posteriori Error Boundsmentioning

confidence: 99%

“…The primary distinction between the proposed T-MLEM method and the methods proposed in Refs. [15,45] is the consideration of recursive regression models that employ latent variables. The MLEM method [15] considered only static problems and thus did not require the use of such variables.…”

Section: Machine-learning Frameworkmentioning

confidence: 99%

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Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Parish

Carlberg

2020

Computer Methods in Applied Mechanics and Engineering

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This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref.[15] to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features-which comprise error indicators that are derived from standard a posteriori error-quantification techniques-to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

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“…wheres n : D → R. As in Ref. [15], we now discuss three types of surrogate models that are often employed to generate the approximate solutionsx n (µ), n = 0, . .…”

Section: Approximate Solutionsmentioning

confidence: 99%

Section: A Posteriori Error Boundsmentioning

confidence: 99%

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Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Parish

Carlberg

2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…Statistical thinking had contributed several aspects of machine learning, for example, in developing computationally intense data classification algorithms, methods in data search and matching probabilities, data mining techniques, model classification and model fitting algorithms, and a combination of all these (see for example, (17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29) and for a collection of articles related to statistical methods in machine learning see (30). Model-based machine learning methods (31) and the construction of coefficients in a regression model can be benefited by machine learning methods (32).…”

Section: Appendix Iii: Machine Learning Versus Deep Learning In Compumentioning

confidence: 99%

“…Statistical and stochastic modeling principles were applied in deep learning algorithms to strengthen the object search capabilities or for improved model fitting in uncertainty (57,59). Boltzmann machines assist in the deep understanding of the data by linking layer level structured data and then by estimating model parameters through maximum likelihood methods (60,61).…”

Section: Appendix Iii: Machine Learning Versus Deep Learning In Compumentioning

confidence: 99%

Deep Learning of Markov Model Based Machines for Determination of Better Treatment Option Decisions for Infertile Women

Rao

Diamond

2019

Preprint

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From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

Qureshi,

Soomro,

Naseem

et al. 2024

Math Methods in App Sciences

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Root‐finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory‐based root‐seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real‐time systems, complicated simulations, and high‐performance computing demand frequent, large‐scale calculations. This article proposes two unique root‐finding methods that increase the convergence order of the classical Newton–Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory‐based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two‐step memory‐based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory‐based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory‐based approaches outperform several existing memory‐based methods when applied to models including a thermistor, path traversed by an electron, sheet‐pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.

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

Cited by 32 publications

References 76 publications

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Deep Learning of Markov Model Based Machines for Determination of Better Treatment Option Decisions for Infertile Women

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

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