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Abstract

A lot of problems involve unknown data relations, which can define a derivative based model of dependent variables generalization. Standard softcomputing methods (as artificial neural networks or fuzzy rules) apply usual absolute interval values of input variables. The new proposed differential polynomial neural network makes use of relative data, which can better describe the character regarding a wider range of input values. It constructs and resolves an unknown partial differential equation, using fractional polynomial sum derivative terms of relative data changes. This method might be applied to solve problems concerned a visual pattern generalization or complex system modeling.

1 Introduction

( 1 ) m – number of variables

A(a1, a2, ... , am), ... - vectors of parameters X(x1, x2, ... , xm) - input vector

D-PNN resulted from the GMDH polynomial neural network (Fig.1.), which was created by a Ukrainian scientist Aleksey Ivakhnenko in 1968 [3]. When the backpropagation technique was not known yet a technique called Group Method of Data Handling (GMDH) was developed for neural network structure design and parameters of polynomials adjustment. General connection between input and output variables is expressed by the Volterra functional series, a discrete analogue of which is Kolmogorov-Gabor polynomial ( 1 ). This polynomial can approximate any stationary random sequence of observations and can be computed by either adaptive methods or system of Gaussian normal equations.

y’ = a0 + a1xi + a2xj + a3xixj + a4xi2 + a5xj2 ( 2 )

GMDH decomposes the complexity of a process into many simpler relationships each described by low order polynomials ( 2 ) for every pair of the input values. Typical GMDH network maps a vector input x to a scalar output y', which is an estimate of the true function f(x) = y. Each neuron of the polynomial network fits its output to the desired value y for each input vector x from the training set. It defines an optimal structure of complex system model with identifying non-linear relations between input and output variables [5]. 2

Differential polynomial neural network The basic idea of the D-PNN is to create and replace a partial differential equation (DE) ( 3 ), which is not known in advance and is able to describe a system of dependent variables, with a sum of fractional multi-parametric polynomial derivative terms ( 4 )[2].

a + ∑i=n1 bi ∂∂xui + ∑i=n1 ∑jn=1 cij ∂x∂i2∂ux j + ... = 0 u = ∑k∞=1 uk u(x1, x2,, … , xn) - searched function of all input variables (dependent variable) a, B(b1, b2,, ..., bn), C(c11, c12, ,... ) - parameters ( 3 )( 4 )

The applied method of integral analogues replaces math operators and symbols of a DE by ratio of corresponding variables. Derivatives are replaced by the integral analogues, i.e. derivative and all operators are replaced by analogous or proportion marks in equations [4].

u i = (a0 + a1 x1 + a 2 x2 + a3 x1 x2 + a 4 x12 + a5 x22 + ...)m n =

b0 + b1 x1 + ... ∂x1∂x2 ...∂xm n – combination degree of a complete polynomial of n-variables m – combination degree of denominator ∂ m f ( x1 ,..., xn ) ( 5 )

The numerator of a term ( 5 ) is a polynomial of all n-input variables of a single neuron and partly defines an unknown function u of eq. ( 3 )( 4 ). The denominator is a derivative part of a DE term ( 5 ), which arose from the partial derivation of the complete n-variable polynomial by competent variable(s). The root function of numerator takes the polynomial into competent combination degree but needn’t be used at all if not necessary.

A block of the D-PNN (Fig.1.) consists of basic neurons, one for each fractional polynomial ( 5 ), defining a sum partial derivative term of the DE ( 3 ) solution. Blocks of higher layers are additionally extended with compound neurons of composite functions, which apply previous layer block outputs and inputs. Each block contains a single output polynomial (without derivative part), thus the block skeleton of the DPNN is formed by the GMDH network. Neurons don’t affect the block output but are applied directly in the sum of a total output calculation of a PDE composition ( 4 ). Each block has 1 and neuron 2 vectors of adjustable parameters a, resp. a, b.

Combination degree n=3 x1 x2 x3

Input variables ( 5 ) Π /

/ Polynomial Block output

Neurons Compound neurons

Root mean square (RMS) error method ( 6 ) was applied for polynomial parameter optimization and PDE term selection.

M ∑ (y d − yi ) i=1

2

M E = → min 3

Multi-layered backward D-PNN

Multi-layered D-PNN forms composite polynomial functions (Fig.2.). Compound DE terms, i.e. derivatives in respect to variables of previous layers, are calculated according to the composite function partial derivation rules ( 7 )( 8 ). They are formed by products of partial derivatives of external and internal functions.

F(x1, x2, … , xn) = f(y1, y2, … , ym) = f(φ1(X), φ2(X),..., φm(X)) i = 1, … , m ∂F ∂xk

m ∂f ( y1, y2 ,..., ym ) ⋅ ∂φ i ( X ) = ∑ i=1 ∂yi ∂xk k=1, … , n

Each block of the D-PNN involves basic neurons e.g. ( 9 ), at first of only linear regression. Additionally blocks of the 2nd and following hidden layers are also extended with neurons, which form composite derivatives utilizing outputs and inputs ( 6 ) ( 7 ) ( 8 ) of back connected previous layer blocks, e.g. the 1st block of the last (3rd) hidden layer (10)(11) [7]. 1 y1 = ∂f ( x∂2x12,1x22 ) = w1 (a0 + a1 x221 ⋅+(ba02 +x2b21+x2a1)3 x21 x22 ) 2 1 y2 = ∂f (x∂2x11,1x22 ) = w2 (a0 + a1x21 + a22⋅xx2222 + a3 x21x22 ) 2 ⋅

The best-fit neuron selection is the initial phase of the DE composition and may apply a proper genetic algorithm (GA). Parameters of polynomials might be adjusted by means of difference evolution algorithm (EA), supplied with sufficient random mutations [1]. The parameter optimization is performed simultaneously with the GA term combination search, where may arise a quantity of local and global error solutions. There would be welcome to apply an adequate gradient descent method too, which parameter updates result from partial derivatives of polynomial DE terms in respect with the single parameters [6]. The number of network hidden layers coincides with a total amount of input variables.

k ∑ yi Y = i=1 k k = amount of active DE terms (11)

Only some of all potential combination DE terms (neurons) may participate in the DE composition, in despite of they have an adjustable term weight (wi). D-PNN’s total output Y is the sum of all active neuron outputs, divided by their amount k (11).

4 Identification of data relations

Consider first only a linear simplification of data relations, thus only linear polynomials of neurons and blocks might be applied. D-PNN consisting of only 1 block of 2 neurons, all terms of the DE (12), is able to identify simple linear 2variable dependence (function), e.g. x1 = 2x2.

y = w1 (a0 + a1 x1 + a2 x2 + a3 x1 x2 ) 12 b0 + b1 x1 + w2 (a0 + a1 x1 + a2 x2 + a3 x1 x2 ) 12

More complicated dependence, where 2 variables depend on a 3rd (e.g. x1 + x2 = x3) may be resolved again D-PNN with one 3-variable combination block. The complete DE (of 1 and 2-combination derivatives) consists of 6 sum terms (neurons) but only 3 may be employed, derivative terms for x3 (13), x1x3 (14), x2x3 (15). Some neurons must be inactivated, having an undesirable effect on the network correct operation. The applied 2-variable combination block D-PNN has 3 hidden layers (Fig.2.).

y1 = w1 (a0 + a1x1 + a2x2 + a3bx03 ++ba14xx31x2 + ...+ a7x1x2x3 )13 y2 = w2 y3 = w3 (a0 + a1x1 + a2x2 + a3x3 + a4x1x2 + ... + a7x1x2x3 )23 b0 + b1x1 + b2x3 + b3x1x3

2 ( a0 + a1x1 + a2x2 + a3x3 + a4x1x2 + ...+ a7x1x2x3) 3

D-PNN can indicate the learned dependence of 3 variables (function) by the output value 1.0 (or any desired). It was trained with only 6 data samples (Tab.1), which were selected to involve proportionally the whole training data interval values <0,500>. However the output function values x3=x1+x2 (x-axis) of the test random input vectors can exceed the maximal trained sum value 500, while the response is (12) (13) (14) (15) kept (Fig.3.). Output errors can result from very disproportional random vector values, which D-PNN was not trained to, e. g. 360 = 358 + 2.

The identification of data relations might be applied to a generalization of fragmented visual patterns into some characteristic dependent elements, which shape assume moved or sized form in the input matrix and where ANN applications fail [7]. The outcomes of 1-block and multi-layered D-PNN are comparable, however the 2nd type is able to involve far larger amount of DE terms and so form a more accurately description of a model.

5 Function approximations

D-PNN can approximate a multi-parametric function, analogously to the ANN approach. Consider the sum function again yt = x1 + x2 + x3, however it could be any linear function. The network with 3 input variables, forming 1 output y = f(x1, x2, x3) should approximate the true function yt by means of sum derivative terms of the partial DE solution. The training data was necessary to be doubled into 12 samples (Tab.2). The D-PNN and ANN approximation is co-equal on the trained interval values <6, 520>, however the ANN approximation ability rapidly falls outside of this range (Fig.4.). The type and operating principle of the D-PNN is the same with applied the dependence identification (Fig.2.), though requiring more time-consuming adjustment.

⎛ ∂u ∂u ∂2u ∂2u ∂2u ⎞

F⎜⎜⎝ x, y,u, ∂x , ∂y , ∂x2 , ∂x∂y , ∂y 2 ⎠⎟⎟ = 0 where F(x, y, u, p, q, r, s, t) is a function of 8 variables (16)

In the case of a real-data application D-PNN processes 2-combination square polynomials of blocks and neurons (DE terms), the same as applied by the GMDH algorithm ( 2 ). This simple polynomial type proves to yield best results besides an easy use and improves also the linear function approximation (which is notable). Thus each block includes 5 basic neurons of derivatives x1, x2, x1x2, x12, x22 of the 2nd order partial DE ( 3 ) of an unknown 2-variable function u, which might be transferred into form of eq. (16). Without this extension only a linear regression of the training data set would be applied. The square and combination derivative terms are also calculated according to the composite function derivation rules (17)(18). However they don’t apply the complete sum of the formulas but only 2 simple terms with 1st order external function derivatives e.g. (19).

F(x, y) = f (u, v) = f [ϕ (x, y),ψ (x, y)]

∂∂v2 2f ⎛⎝⎜ ∂∂ψx ⎞⎟⎠ 2 + ∂∂uf ⋅ ∂∂x2ϕ2 + ∂∂fv ⋅ ∂∂x2ψ2 y4 = ∂2 f (∂xx21211, x22) = w4 (a0 + a1x21 + a2x22 +3a⋅3xx2212x22 + a4x221 + a5x222)12 ⋅ (b) (c) (d)

The 3-variable D-PNN applying extended polynomials ( 2 ) of blocks, neurons and square, combination DE terms, tried to predict the static pressure of 1 site locality time-series (Fig.5a-d). It was trained along with the 1-layer ANN the previous day hourly pressure data series (24 or 48 hours, i.e. data samples), which are free on-line available [9]. Meteorological forecasts require as a rule high amount of state input variables to define a complex model, however some tendencies of the progress curves are notable. The D-PNN 2 models double the applied amount of neurons compared with D-PNN 1 (Fig.5.). The DE composition based predictions of the D-PNN seems to succeed any better. The more varied models are formed than ANN (applying 4 or 5 input variables), which is induced by a different neuron combination selection. 6

Conclusion

D-PNN is a new neural network type, which identification and function approximation is based on generalization of data relations. The relative data processing is contrary to common soft-computing method approachs (e.g. ANN), which applications are subjected to a fixed interval of absolute values. This handicap disallows to use various learning and testing data range values (Fig.4.), which may involve real data applications. Thus D-PNN’s non-linear regression can cover a generalization of wider interval values. It forms and resolves a DE, composed of sum fractional derivative terms, defining a system model of dependent variables. It is trained only with a small set of input-output data samples, likewise the GMDH algorithm does [1]. The inaccuracies of presented experiments can result from applied incomplete rough training and selective methods, requiring large improvements. Behavior of the presented method differs essentially from other common neural network techniques.

Acknowledgement

This work has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme ’Research and Development for Innovations’ funded by Structural Funds of the European Union and by the Ministry of Industry and Trade of the Czech Republic, under the grant no. FR-TI1/420 , and by SGS, VŠB – Technical University of Ostrava, Czech Republic, under the grant No. SP2012/58.

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