Applying a Neural Network Collocation Method to an Incompletely Known Dynamical System via Weak Constraint Data Assimilation

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Abstract
A method based on a neural network collocation method is proposed for approximating incompletely known dynamical systems via weak constraint data assimilation formulation. The aim of the new method is to solve several difficult issues encountered in previous research. For this purpose, the weak constraint property of the neural network collocation method is used. The problem regarding the wider assimilation window is tackled by interconnecting narrower windows with finite overlapping interfaces. The method is examined by considering the Lorenz system as an example where one of the three equations of the system is unknown. The object function of the neural network training is composed of squared residuals of differential equations at collocation points and squared deviations of the observations from their corresponding calculated values. The weakly and highly nonlinear cases of the Lorenz system are considered. The numerical experiments have been carried out with simulated noiseless and noisy observation data under various conditions. The performance of the method for approximating an unknown equation during the assimilation and testing periods is examined for the two cases. Also, the parameters of incomplete dynamical systems are estimated for the two cases. Satisfactory results have been obtained in both cases.