On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions

• URL: http://arxiv.org/abs/2411.12940v1
• Date: Wed, 20 Nov 2024 00:18:46 GMT
• Title: On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions
• Authors: Jake Buzhardt, C. Ricardo Constante-Amores, Michael D. Graham,
• Abstract summary: We show that extended dynamic mode decomposition with dictionary learning (EDMD-DL) is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow.
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• Abstract: This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time-evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary learning (EDMD-DL), when combined with a state space projection, is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We highlight how this projection step introduces nonlinearity into the evolution equations, enabling significantly improved EDMD-DL predictions. With this projection, EDMD-DL leads to a nonlinear dynamical system on the state space, which can be represented in either discrete or continuous time. This system has a natural structure for neural networks, where the state is first expanded into a high dimensional feature space followed by a linear mapping which represents the discrete-time map or the vector field as a linear combination of these features. Inspired by these observations, we implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow, showing comparable performance across methods in terms of short-time trajectory prediction, reconstruction of long-time statistics, and prediction of rare events. We also show that these methods provide comparable performance to a non-Markovian approach in terms of prediction of extreme events.

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