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Hierarchical deep learning of multiscale differential equation time-steppers.
- Source :
-
Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences . 8/8/2022, Vol. 380 Issue 2229, p1-17. 17p. - Publication Year :
- 2022
-
Abstract
- Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration expensive. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the dynamical system flow map over a range of time-scales. The model is purely data-driven, enabling accurate and efficient numerical integration and forecasting. Similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our hierarchical time-stepping scheme provides advantages over current time-stepping algorithms, including (i) capturing a range of timescales, (ii) improved accuracy in comparison with leading neural network architectures, (iii) efficiency in long-time forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on numerous nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto–Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing and clockwork RNN. This article is part of the theme issue 'Data-driven prediction in dynamical systems'. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 1364503X
- Volume :
- 380
- Issue :
- 2229
- Database :
- Academic Search Index
- Journal :
- Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences
- Publication Type :
- Academic Journal
- Accession number :
- 157933593
- Full Text :
- https://doi.org/10.1098/rsta.2021.0200