We present a method for training a deep neural network containing sinusoidal activation functions to fit to time-series data. Weights are initialized using a fast Fourier transform, then trained with regularization to improve generalization. A simple dynamic parameter tuning method is employed to adjust both the learning rate and regularization term, such that stability and efficient training are both achieved. We show how deeper layers can be utilized to model the observed sequence using a sparser set of sinusoid units, and how non-uniform regularization can improve generalization by promoting the shifting of weight toward simpler units. The method is demonstrated with time-series problems to show that it leads to effective extrapolation of nonlinear trends.
A more sophisticated group of methods involves neural networks with recurrent connections @cite_34 . These produce their own internal representation of state. (See Figure .B.) This enables them to learn how much to adjust their representations of state based on observed values, and hence operate in a manner more robust against noisy observations.
@cite_34
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The paper first summarizes a general approach to the training of recurrent neural networks by gradient-based algorithms, which leads to the introduction of four families of training algorithms. Because of the variety of possibilities thus available to the "neural network designer," the choice of the appropriate algorithm to solve a given problem becomes critical. We show that, in the case of process modeling, this choice depends on how noise interferes with the process to be modeled; this is evidenced by three examples of modeling of dynamical processes, where the detrimental effect of inappropriate training algorithms on the prediction error made by the network is clearly demonstrated. >