JPH07192072A - Physical quantity calculation device relating to glass and ceramics - Google Patents

Abstract

PURPOSE:To calculate continuous physical quantities regardless of the kinds of output and composition materials, to be applicable even when the physical quantity is nonlinear and to simplify algorithm. CONSTITUTION:This device is provided with a hierarchical neural network 4 composed of an input layer 1 which is the layer of elements for inputting the composition ratio of the respective component elements of the composition materials of glass and ceramics and production conditions as input values, an output layer 3 which is the layer of the elements for taking out the physical quantity relating to the glass and the ceramics as an output value and an intermediate layer 2 which is the layer of the elements other than the input layer and the output layer and a learning means 5 for inputting the composition ratio of the respective component elements of the composition material of the known glass and ceramics and the production conditions to the respective elements of the input layer 1, supplying the physical quantity corresponding to the input value and performing the learning of the hierarchical neural network 4 and the physical quantity relating to the unknown glass and ceramics is obtained.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION The present invention is based on data such as glass and ceramics composition and the like.
Regarding a device for predicting and calculating physical quantities such as a lattice constant and a dielectric constant, for example, predicting a composition ratio for obtaining target glass and ceramics, predicting a glass transition point of a condition necessary for press forming, and the like The present invention relates to a physical quantity calculation device applied to composition design.

[0002]

2. Description of the Related Art There has been proposed a method of relating a glass composition ratio and a physical quantity such as a refractive index, which has been previously known by experiments, by using the addition rule of [Equation 1].

[0003]

[Equation 1] Where a is a physical quantity, p _i is the percentage of metal oxide _i in glass, x _i is a coefficient factor of metal oxide _i in glass, and n is
Is the number of components.

Utilizing this addition rule, for example, AAAppe
nn (Glass Chemistry, Nisso News Agency (1984)), MLHuggins
(J.Opt.Soc.Amer., 30,420 (1940), etc.)
A method of calculating a physical quantity such as a refractive index and a dispersion of an oxide glass containing iO ₂ as a main component has been reported. A glass material design support device that integrates these various calculation methods is proposed in Japanese Patent Laid-Open No. 2-111642.

The addition rule shown in the equation (1) is, for example, 2
When the glass component system, component as ideally FIG 2 _(a) X a O _b and component Y _c O components by the composition ratio of _d X _a
It is shown as a straight line connecting the physical quantity of O _{b and} the physical quantity of the component Y _c O _d . However, in reality, as shown in FIG. _2B , there is a case where a proportional relationship does not hold between the composition ratio of the components X _a O _b and Y _c O _d and the physical quantity. In such a case, a constant value cannot be used for the coefficient x _i . Therefore, the composition range is divided and the coefficient x _i is changed for each,
A method such as facilitating approximation with a straight line or changing the coefficient x _i depending on the component contained in the glass has been adopted.

Further, a method for calculating the physical quantity of a fluoride glass by using a linear regression equation and an additive law is described in the 32nd glass and photonics material discussion meeting of the Ceramic Society of Japan (1991) p. 55, Bulletin of the Japan Institute of Metals 31
[7] (1992) p. Proposed in 604. When the functional relationship between the input data and the output data is approximated as in this method, this method can be applied to calculations for glass composition substances other than oxides and other physical quantities. Further, JP-A-5-54162 discloses a method of inputting a composition ratio and determining whether vitrification is possible.

[0007]

However, by using the additive rule shown in the formula of [Equation 1], AAAppenn (Glass Chemistry, Nisso Press (1984)), MLHuggins (J.Opt.Soc .Ame
r., 30,420 (1940), etc.) and Japanese Patent Laid-Open No. 2-11164.
The method for calculating physical quantities such as the refractive index and dispersion of glass by the glass material design support apparatus proposed in Japanese Patent No. ₂ is limited to oxide glass whose substance is SiO ₂ as a main component, and its constituent components. Are also limited to specific oxides. Moreover, the content is also limited, and the physical quantity that can be calculated is also limited. For example, the glass transition point is an important physical quantity in manufacturing a press lens in which glass is molded by a mold, but cannot be calculated.

In addition, according to the addition rule shown in the equation (1),
Actually, as shown in FIG. 2B, the components X _a O _b and Y _c O _{d are}
If there is no proportional relationship between the composition ratio and the physical quantity of
Divide the composition range and change the coefficient x _i for each,
Since the curve was approximated by changing the coefficient x _i depending on the component contained in the glass, the algorithm became complicated and the calculation accuracy was not very good.

Proceedings of 32nd Symposium on Glass and Photonics Materials, Japan Ceramic Society (1991) p. 55, Bulletin of the Japan Institute of Metals 31 [7] (1992) p. The method of calculating the physical quantity of a fluoride glass by using the linear regression equation and the addition rule proposed in 604 is applied to glass composition substances other than oxide glass mainly composed of SiO ₂ and other physical quantities. Can also be applied, but is effective only when additivity is established and the number of data is large. When the number of data is small, the accuracy becomes extremely poor.

Generally, a glass composition such as a fluoride-based, chalcogenide-based, or telluride-based glass has a new history as compared with an oxide-based glass containing SiO ₂ as a main component. Therefore, the type and amount of data may not be sufficient, and the accuracy of the physical quantity calculated at this time was not sufficient. Further, in a certain type of glass system, there is a phenomenon (abnormal phenomenon) in which the property remarkably deviates from the additivity and a marked maximum, minimum, or abrupt gradient change occurs on the composition-dependent curve of the substance (physical quantity). For example, it is called a boric acid abnormality or a mixed alkali effect. In such a case, it is necessary to use a plurality of coefficients x _{i in} the equation of [Equation 1] for dividing the composition range, and for this reason, it is necessary to perform linear regression several times, and the processing is very difficult. It becomes complicated. Moreover, since the data is divided and used for linear regression, there is a problem that the number of data is further reduced and the accuracy is lowered. Therefore, even if the target is limited to a specific composition substance and a specific output (for example, the refractive index) and the algorithm is complicated, the improvement in accuracy cannot be expected very much. Moreover, if the glass composition further increases in the glass composition exhibiting the abnormal phenomenon of boric acid, which is an abnormal phenomenon, and the mixed alkalinity effect, it cannot be applied at all. In addition, many ceramics do not follow the additivity, and it was not possible to calculate the physical quantity using the additivity. Here, the precision means the relation between the input data (composition ratio, etc.) given in advance and the output data (physical quantity such as refractive index), and the output when the input data is input again. It is the difference from the desired output (for example, squared error).

Also, in the method of determining the vitrification by inputting the composition ratio in Japanese Patent Laid-Open No. 5-54162, the output value is a discrete value of 1 when vitrification occurs and 0 when not vitrification. Therefore, this method is equivalent to performing pattern separation. When pattern classification is performed to predict the viability of vitrification, the output can be discrete values, but if you want to output continuous physical quantities, this method cannot be applied. The design could not be done. Moreover, the presence or absence of vitrification can be easily determined experimentally by melting and cooling a small amount of glass raw material.

As described above, according to the conventional technique, the physical quantity cannot be calculated only by pattern classification, or even if the physical quantity can be calculated, the kind of the physical quantity and the glass composition are limited. This method cannot be applied to cases where linearity is not exhibited, and sufficient accuracy was not obtained for calculated physical quantities when the number of data was small.

An object of the present invention is to calculate the physical quantity of a known data with high accuracy of input / output relation enough to interpolate unknown data even if the number of data is small, and to calculate a continuous physical quantity regardless of the type of composition substance or output. It is also possible when the physical quantity does not show linearity in the glass and ceramic composition,
Moreover, it is to provide a physical quantity calculation device for glass and ceramics with a simple algorithm.

[0014]

Therefore, the physical quantity calculating apparatus for glass and ceramics of the present invention inputs the composition ratio of each component element of the composition material of glass and ceramics and the manufacturing conditions as input values as shown in FIG. An input layer 1 that is a layer of an element, an output layer 3 that is a layer of an element that extracts a physical quantity related to glass and ceramics as an output value, and an intermediate layer 2 that is a layer of the element other than the input layer and the output layer. The hierarchical neural network 4 is provided, and the composition ratios of the constituent elements of the known glass and ceramic composition materials and the production conditions are input to each element of the input layer 1 and the physical quantity corresponding to the input value is used as an output value. The learning means 5 is provided to perform learning of the hierarchical neural network 4, and the hierarchical model that has been learned by the learning means 5 is provided. The input layer of the neural network 4 1
The composition ratio of each constituent element of the composition material of unknown glass and ceramics and the production conditions are input to each element of (1) to obtain the physical quantity of the unknown glass and ceramics from the output layer 3. Is.

[0015]

As shown in FIG. 3, the present invention comprises a hierarchical neural network composed of an input layer, an intermediate layer and an output layer. First, the structure and operation of the hierarchical neural network will be described. Elements that perform signal processing exist in the input layer, the intermediate layer, and the output layer. Generally, a layer of an element that inputs an input value is called an input layer, a layer of an element that extracts an output value is called an output layer, and a layer of the other elements is called an intermediate layer. The elements of the output layer and the intermediate layer are respectively coupled to the elements of the previous layer, and a coupling weight value called weight is assigned to each coupling. A value obtained by weighting the output of each element of the previous layer is input to the elements of the output layer and the intermediate layer, and the sum is calculated. In some cases, a bias term having a value unique to the element may be added to the sum. The result of applying a function called an input / output function to the obtained sum value becomes the output value of the element.

Specifically, for example, the input layer element is represented by i, the intermediate layer element is represented by j, the output layer element is represented by k, the element input is represented by u, and the output is represented by x. u _j , the output thereof is x _j , the output of the element i of the input layer is x _i , the weight (coupling weight value) of the element j of the intermediate layer and the element i of the input layer is W _ji , and the bias of the element j is θ _j. , And the input / output function is G (x), the relationship between the input layer and the intermediate layer is described by the equations [2] and [3].

[0017]

[Equation 2]

[0018]

[Equation 3]

The relationship between the intermediate layer and the output layer, or when there are a plurality of intermediate layers, the relationship between the intermediate layers before and after is also [Equation 2].
It is shown by the same relational expression as the expression of. Here, the output of this network can be made a continuous value by selecting a function that outputs a continuous value with the input / output function in the output layer being G (x).

By taking various values of the weight and the bias in this network, it is possible to approximate the desired mapping relationship.

Updating the weight and bias values so that a desired output is obtained for a certain input is called learning of a neural network. As a learning method, a back-propagation method that minimizes a squared error between a desired output (teaching data) for a certain input and an actual network output may be used. The back propagation method will be described below.

It is assumed that the k element is the element of the output layer, the j element is the element of the intermediate layer, and the elements are coupled in the relationship of the equations [2] and [3]. For a given input signal, letting t _k be the desired output that the k element should produce and x _{k the} actual output,
The backpropagation method is an algorithm that changes the network weight and bias value so that [Equation 4] is minimized.

Here, in particular, updating the weight value is considered. (The updating of the bias value is also considered.)

[0024]

[Equation 4]

Since x _k can be expressed as a function of the weight value, E
Forms a curved surface in space with the weight value as the coordinate axis. In order to obtain the minimum value of this E, the value of the weight is updated according to the equation of [Equation 5] according to the idea of the steepest descent method.
Note that ε is a positive constant.

[0026]

[Equation 5]

The calculation of the equation (5) is as follows.

[0028]

[Equation 6]

That is, the weight w _ji of the k element may be updated according to the equation (6). Using the formula of [Formula 4], the formula of [Formula 6] can be expressed as the formula of [Formula 7].

[0030]

[Equation 7]

With respect to a certain input signal, the weight value of the output layer is updated from the desired output and the actual output as in the equation (7).

When the j-th element is the intermediate layer, the value of ∂E / ∂x _j is calculated from the k-element of the layer after the j-th element layer as in the equation (9). However, it is assumed that the j layer and the k layer are also connected in the same relationship as the expressions of [Equation 2] and [Equation 3].

In the case of the input layer and the intermediate layer, the update of W _ji is expressed by the equation [8]. However, i layer and j layer are
It is assumed that they are connected by the relationship of the expressions [Formula 2] and [Formula 3].

[0034]

[Equation 8]

Here, ∂E / ∂x as in the equation (9)
The value of _j is calculated from the k elements of the output layer of the rear layer from the intermediate layer of the j element.

[0036]

[Equation 9]

The result of the equation (9) is substituted into the equation (6) to update the weight value of the intermediate layer.

This method in which the weight values are updated in the reverse order in order from the output layer to the intermediate layer is the back propagation method.

The updating of the bias value is also performed by the equation (5), and the updating rule can be similarly derived by differentiating E with respect to θ _j , and the details thereof will be omitted.

By combining the elements in which the values of the weight and the bias are appropriately selected by learning, the above hierarchical network can be accurately approximated even with a non-linear mapping relationship. Further, the configuration is a simple combination of elements having an input / output relation as shown by the equations [2] and [3], and the algorithm is very simple.

Therefore, the physical quantity calculating apparatus for glass and ceramics of the present invention sets the weight and the bias value by learning the known data by using this hierarchical neural network, and then the unknown data such as glass and ceramic composition data. Is used to approximate a function for obtaining a physical quantity such as a refractive index, dispersion, glass transition point, expansion coefficient and softening point. In this case, the composition ratio and the manufacturing conditions are input to the input layer, and the physical quantity is extracted from the output layer. Here, the production conditions include, for example, glass and ceramics production conditions, physical quantity measurement conditions, coordination number of metal ions in glass and ceramics, ionic radius, bond energy between atoms constituting glass and ceramics, and other conditions. Say.

The number of components of the input value becomes the number of elements in the input layer,
The number of types of physical quantities taken out as output is the number of elements in the output layer. For example, when one refractive index is obtained from the ternary glass composition of SiO ₂ , K ₂ O and PbO, the number of elements in the output layer may be 3 and the number of elements in the output layer may be 1. The number of intermediate layers and the number of elements may be arbitrarily selected.

In order to output the physical quantity of continuous values, the input / output function of the output layer must use the function of outputting continuous values. However, in the middle layer, it is sufficient that at least one input / output function is continuous, and it is even better if all input / output functions are continuous.

The neural network thus constructed is made to learn the relationship between the input data and the output data which is known in advance. This learning is performed until the input-output mapping relationship is sufficiently approximated.

FIG. 5 shows an algorithm for learning n sets of input and teacher data, which summarizes the above. In the learning process, first, the weight value and the bias value of the intermediate layer and the output layer are initialized, and then n sets of input data are set to calculate the outputs of the elements of the intermediate layer and the output layer to obtain the actual output and the desired output. Calculate the difference from the output. Further, the calculation of the error differential value and the update of the weight value and the bias value are sequentially performed for the output layer and the intermediate layer. After performing this for n sets of input data, the sum of the squared errors of the n sets is calculated to determine whether it is within the allowable error range. The above processing is repeated until the sum of the squared errors falls within the allowable error range, and when the error falls within the allowable error range, the learning ends.

An output obtained when an input whose output is not known in advance (unlearned data) is put into the network after the above learning is performed becomes a predicted output for the input. For example, let the network learn a known set of glass and refractive index data. Next, the glass composition whose refractive index is unknown is input, and the output obtained as a result becomes the predicted value of the unknown refractive index. An algorithm for predicting this physical quantity is shown in FIG. In this prediction process, input data whose output value is unknown (composition ratio, manufacturing conditions, etc.) is set in each element of the input layer, and the output calculation of the intermediate layer and output layer is performed to obtain the output value from the output layer. Prediction values for unknown input data are obtained.

[0047]

EXAMPLES Next, the present invention will be described in more detail with reference to examples. FIG. 4 shows a structural diagram of the first embodiment of the present invention.
The present embodiment is a three-layer neural network with one input layer, one intermediate layer, and one output layer, and the number of elements in each layer is 3, 30, and 1, respectively. SiO ₂ and K as input
_For the composition ratio of ₂ O and PbO and the output, the Abbe number is selected.
The input / output function of the intermediate layer uses a sigmoid function given by the following equation (10), and the output layer uses a linear function of the equation (11).

[0048]

## EQU10 ## G (u) = 1 / (1 + exp (-u))

[0049]

[Equation 11] G (u) = u Using the illustrated input values, output values, weights, etc.,

[0050]

For the element 101 of the intermediate layer, u ₁₀₁ = W _101.1 x ₁ + W _101.2 x ₂ + W _101.3 x ₃ + θ ₁₀₁ x ₁₀₁ = G (u ₁₀₁ ) = 1 / (1 + exp (-u ₁₀₁ )) output layer For 201, u ₁₀₁ = W _201.101 x ₁₀₁ + W _201.102 x ₁₀₂ + W _201.103 x ₁₀₃ ...... + W _201.130 x ₁₃₀ + θ ₂₀₁ output x ₂₀₁ = G (u ₂₀₁ ) = u ₂₀₁ teacher data t ₂₀₁ E = (x ₂₀₁ -t ₂₀₁₎ ^2/2 so that the minimum W _101.1, W _101.2, ...... and W
_201.101 , ..., Update W _201.102 .

Here, the input / output function in the output layer is a function that takes continuous values.

After 42 sets of inputs and their teaching data are given to the present network for learning, in order to examine the mapping approximation accuracy, the output for all 42 inputs and the desired output, the teaching data, are used. The sum of squared errors was calculated. The results are shown in Table 1 in comparison with the conventional method. Also,
Of the 42 sets of data, only 21 sets are arbitrarily selected for learning, the remaining 21 sets are input as unlearned data, and the same sum of squared errors is calculated by combining the learned and unlearned data. Then, the data interpolation ability was investigated. The results are also shown in Table 1.

[0053]

[Table 1] It is clear from Table 1 that the accuracy of the mapping approximation of this embodiment is much better than that of the conventional method. Further, even when unlearned data is input, its accuracy is better than that of the conventional method using all data, so it can be said that it has excellent interpolation ability.

In this embodiment, the input / output function is selected as in the equations (10) and (11), but any function that outputs a continuous value can be used in the output layer. Further, although the intermediate layer may use an arbitrary function, particularly when a non-linear function is used, it can more effectively act on a non-linear function approximation problem.

Now, in order to reduce the dependency of the initial weight value on the learning result, the initial value of the weight is set so that the difference between the output with respect to the input and the teacher data is reduced as much as possible in the initial state before learning. (Usually a random value)
Alternatively, the bias value may be adjusted or the data may be standardized. Specifically, the element j having the output layer of the present embodiment
Will be described with reference to the relationship between the element group and the element group of the intermediate layer.

X _j is the output of the output layer j element, X _i is the output of the intermediate layer i element, and W _ji is the output layer element j and the intermediate layer element i.
And the weight (coupling constant), θ _j is the bias of element i, G
If (u) is an input / output function, an expression like [Equation 13] is obtained.

[0057]

[Equation 13]

[X] indicates the order of x. For example, if the initial value of the weight or the bias value is adjusted so as to satisfy the following equation (14) or the data is standardized, the value before learning In the initial state, the difference between the output with respect to the input and the teacher data is reduced.

[0059]

[Equation 14] Also in the case of the relationship between the input layer and the intermediate layer, in the formula of [Equation 14], [teaching data] is [average value of X _j ], [average value of X _i ] is [input data], [element of intermediate layer] The same relational expression can be obtained by replacing [number] with [the number of elements in the input layer] and [initial value of W _ji ] with [initial value of weights of the input layer and the intermediate layer]. From the above discussion, in the present embodiment, the dependence of various values on the initial value of the learning result is reduced by using a constant such as the expression of [Equation 15] or normalizing the data.

[0060]

[Equation 15] −0.03 ≦ w _ji initial value ≦ 0.03 (common to all layers) −0.03 ≦ initial bias value ≦ 0.03 (common to all layers) 0 ≦ teaching data ≦ 1.0 in average = 0.5 second embodiment] the first embodiment of 0 ≦ input data ≦ 10.0 X _i, the physical quantity to be output is was only one kind of "Abbe number", to the output When the number of physical quantities to be used is two or more, a plurality of physical quantities can be taken out at the same time if the number of elements in the output layer is made plural, so that the man-hour can be reduced when calculating the plurality of physical quantities. For example, in addition to the "Abbe number" of Example 1,
In order to take out two kinds of physical quantities of “refractive index” and “thermal expansion coefficient”, the number of elements in the output layer is set to three.

[Third Embodiment] In the first embodiment, a cooling temperature is applied to the input in addition to the ceramic composition. Even if ceramics have the same composition, the physical quantity (for example, electric conductivity) may differ depending on the firing conditions, and the present invention is also effective when predicting the physical quantity in such a case.

[Fourth Embodiment] In the first embodiment, the input data is the component ratio of Ge and the alkali metal, and the output data is the refractive index. The relationship between the input and output data shows a non-linearity in which the composition dependence curve of the physical quantity largely deviates from the addition rule, and the present invention effectively works even in such a case.

The present invention is not limited to the above embodiment, but various modifications can be made. In particular, since it does not depend on the type of input / output data, the problem to be applied is not limited to oxide-based glass, and for example, fluoride-based glass, chalcogenide-based glass, telluride-based glass, oxynitride glass, oxyfluoride glass. Any composition system such as, for example, sulphate glass, etc. is also effective for composition systems in which the physical quantity is not linear with respect to the glass and ceramic compositions. Further, as the input data, the composition ratio of glass and ceramics, the production conditions, the ionic radius, the binding energy, the coordination number, etc. can be arbitrarily selected.

[0064]

As described above, according to the physical quantity calculation apparatus for glass and ceramics of the present invention, the accuracy of the input / output relationship of known data is high enough to interpolate unknown data even if the number of data is small, and , Continuous physical quantities can be calculated regardless of the type of composition substance and output. Also,
The above-mentioned hierarchical neural network according to the present invention is
Data such as glass and ceramic composition and refractive index,
It is possible to approximate the mapping relationship with the physical quantity of continuous value such as dispersion and glass transition point, and it becomes possible to predict various physical quantities from unlearned data. In addition, the present invention can be applied even when the physical quantity does not show linearity in the glass and ceramic composition, and can realize a physical quantity calculation device for glass and ceramics with a simple algorithm.

[Brief description of drawings]

FIG. 1 is a diagram showing one embodiment of a physical quantity calculation device for glass and ceramics according to the present invention.

FIG. 2 is a diagram showing a relationship between a composition ratio and a physical quantity.

FIG. 3 is a diagram showing a hierarchical neural network which is a main configuration of the present invention.

FIG. 4 is a diagram showing a configuration of a first exemplary embodiment of the present invention.

FIG. 5 is a diagram for explaining a learning algorithm of the present invention.

FIG. 6 is a diagram for explaining a prediction algorithm of the present invention.

[Explanation of symbols]

1 ... Input layer, 2 ... Intermediate layer, 3 ... Output layer, 4 ... Hierarchical neural network, 5 ... Learning means

Claims (1)

[Claims] 1. An input layer which is a layer of an element for inputting a composition ratio of respective constituent elements of glass and ceramic composition substances and manufacturing conditions as input values, and an element layer for extracting a physical quantity relating to glass and ceramics as an output value. A layered neural network consisting of an output layer and an intermediate layer which is a layer of elements other than the input layer and the output layer is provided, and the composition ratio of each component element of the known glass and ceramic composition materials and the production conditions are set. The learning means includes a learning unit that inputs each element of the input layer and gives a physical quantity corresponding to the input value as an output value to learn the hierarchical neural network. For each element of the input layer, the composition ratio of each component element of the composition material of unknown glass and ceramics and the manufacturing conditions are set. A physical quantity calculating device for glass and ceramics, which is configured to input and obtain physical quantities for unknown glass and ceramics from the output layer.