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12 第四届中国运筹青年论坛 | 青年报告

Fractional-order Global Optimal BackPropagation Machine

Trained by Improved Fractional-order Steepest Descent Method

王健 中国石油大学（华东）

This paper introduces a novel fractional-order branch of the family of BackPropagation Neural

Networks (BPNNs) trained by the improved Fractional-order Steepest Descent Method (FSDM); this
differs from the majority of the previous classic first-order BPNNs and as such trained by the
traditional first-order steepest descent method. To improve the optimization performance of classic
first-order BPNNs, in this paper we study whether it could be possible to apply improved FSDM
based on fractional calculus to generalize classic first-order BPNNs to the Fractional-order
Backpropagation Neural Networks (FBPNNs). Motivated by this inspiration, this paper proposes a

state-of-the-art application of fractional calculus to implement a FBPNN trained by an improved
FSDM whose reverse incremental search is in the negative directions of the approximate fractional-
order partial derivatives of the square error. At first, the theoretical concept of a FBPNN trained by
an improved FSDM is described mathematically. Then, the mathematical proof of the fractional-
order global optimal convergence, an assumption of the structure, and the fractional-order multi-
scale global optimization of a FBPNN trained by an improved FSDM are analysed in detail. Finally,
we perform comparative experiments and compare a FBPNN trained by an improved FSDM with a

classic first-order BPNN, i.e., an example function approximation, fractional-order multi-scale
global optimization, and two comparative performances with real data. The more efficient optimal
searching capability of the fractional-order multi-scale global optimization of a FBPNN trained by
an improved FSDM to determine the global optimal solution is the major advantage being superior
to a classic first-order B.