Neural Ordinary Differential Equations for Model Order Reduction of Stiff Systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912434373001216 |
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| author | Caldana, Matteo Hesthaven, Jan S. |
| author_facet | Caldana, Matteo Hesthaven, Jan S. |
| contents | Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, a condition where rapid variations in some components of the solution demand prohibitively small time steps for explicit solvers. This work addresses the stiffness issue when employing neural ODEs for model order reduction by introducing a suitable reparametrization in time. The considered map is data-driven and it is induced by the adaptive time-stepping of an implicit solver on a reference solution. We show the map produces a nonstiff system that can be cheaply solved with an explicit time integration scheme. The original, stiff, time dynamic is recovered by means of a map learnt by a neural network that connects the state space to the time reparametrization. We validate our method through extensive experiments, demonstrating improvements in efficiency for the neural ODE inference while maintaining robustness and accuracy. The neural network model also showcases good generalization properties for times beyond the training data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neural Ordinary Differential Equations for Model Order Reduction of Stiff Systems Caldana, Matteo Hesthaven, Jan S. Numerical Analysis 65L99, 68T07 Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, a condition where rapid variations in some components of the solution demand prohibitively small time steps for explicit solvers. This work addresses the stiffness issue when employing neural ODEs for model order reduction by introducing a suitable reparametrization in time. The considered map is data-driven and it is induced by the adaptive time-stepping of an implicit solver on a reference solution. We show the map produces a nonstiff system that can be cheaply solved with an explicit time integration scheme. The original, stiff, time dynamic is recovered by means of a map learnt by a neural network that connects the state space to the time reparametrization. We validate our method through extensive experiments, demonstrating improvements in efficiency for the neural ODE inference while maintaining robustness and accuracy. The neural network model also showcases good generalization properties for times beyond the training data. |
| title | Neural Ordinary Differential Equations for Model Order Reduction of Stiff Systems |
| topic | Numerical Analysis 65L99, 68T07 |
| url | https://arxiv.org/abs/2408.06073 |