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

Abstract

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.

arXiv comment51 pages

References in corpus (6)

• Adam: A Method for Stochastic Optimization
• Scikit-learn: Machine Learning in Python
• A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions
• The ROMES method for statistical modeling of reduced-order-model error
• Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations
• Statistical closure modeling for reduced-order models of stationary systems by the ROMES method

Cited by in corpus (9)

• Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics
• Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems
• Physics-aware registration based auto-encoder for convection dominated PDEs
• Variational Autoencoders for Learning Nonlinear Dynamics of Physical Systems
• Reduced Order Modeling with Shallow Recurrent Decoder Networks
• Projection-tree reduced order modeling for fast N-body computations
• Generalization capabilities and robustness of hybrid models grounded in physics compared to purely deep learning models
• Hierarchical deep learning-based adaptive time-stepping scheme for multiscale simulations
• Kinematically consistent recurrent neural networks for learning inverse problems in wave propagation