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Time-series machine-learning error models for approximate solutions to parameterized dynamical systems. (English) Zbl 1442.65446
Summary: 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 [B. A. Freno and the second author, ibid. 348, 250–296 (2019; Zbl 1440.65058)] 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.
MSC:
65P99 Numerical problems in dynamical systems
70-08 Computational methods for problems pertaining to mechanics of particles and systems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H30 Classification and discrimination; cluster analysis (statistical aspects)
Software:
Adam; Keras; Scikit
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