Due to the increasing availability of large-scale observation and simulation datasets, data-driven representations arise as efficient and relevant computation representations of dynamical systems for a wide range of applications, where model-driven models based on ordinary differential equation remain the state-of-the-art approaches. In this work, we investigate neural networks (NN) as physically-sound data-driven representations of such systems. Reinterpreting Runge-Kutta methods as graphical models, we consider a residual NN architecture and introduce bilinear layers to embed non-linearities which are intrinsic features of dynamical systems. From numerical experiments for classic dynamical systems, we demonstrate the relevance of the proposed NN-based architecture both in terms of forecasting performance and model identification.
In this work, we investigate neural network (NN) representations for dynamical systems governed by some underlying but unknown ODEs. We aim to derive computationally-efficient and physically-sound representations. Our contribution is three-fold: i) we make explicit the interpretation of Runge-Kutta integration schemes as graphical models to introduce a residual NN architecture, ii) we introduce a NN architecture with bilinear layers to embed intrinsic non-linearities depicted by the dynamical systems, iii) we demonstrate the relevance of the proposed NN architecture with respect to state-of-the-art models both for model identification and forecasting for different classic systems, namely Lorenz-63 and Lorenz-96 dynamics @cite_9 , which are representative of ocean-atmosphere dynamics, and Oregenator system @cite_4 , which relates to oscillatory chemical dynamics.
@cite_9 @cite_4
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Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. The field of oscillating chemical reactions has grown dramatically over the last 15 years. The reasons for this activity are the intrinsic chemical interest in the behavior and mechanisms of oscillating chemical reactions and the connection between oscillating reactions and the properties of nonlinear differential equations.