Method for constructing composite response surfaces by combining neural networks with polynominal interpolation or estimation techniques
First Claim
1. A method for constructing a composite response surface based on neural networks and selected functions, the method comprising providing a computer that is programmed:
- (1) to provide a set of h initial parameters that determine variation of provided data for a target variable, where each parameter corresponds to a coordinate in an h-dimensional parameter space G;
(2) to decompose the h parameters into a first set of s simple parameters fi, numbered i=1, . . . , s, that may be used to describe the provided data with polynomials of total degree no greater than a selected number Ms, and a second set of c complex parameters gj, numbered j=1, . . . , c, that may be used to describe the provided data using neural networks, and with s+c=h, where s, c and Ms are selected positive integers;
(3) to provide a simplex, having s+1 vertices, numbered k=1, . . . , s+1, and centered at a selected point in the space G;
(4) to apply a neural network for each of the s+1 vertices, and to train each of the s+1 neural networks, using selected simulation data obtained by varying the parameters gj to generate a first sequence of network functions Rk(g1, . . . , gc);
(5) to provide a second sequence of shape functions Pk(f1, . . . , fs) that satisfy the conditions Pk(f1, . . . , fs)=1 at the vertex numbered k and Pk(f1, . . . , fs)=0 at any vertex other than vertex number k, and Σ
Pk(f1, . . . , fs)=1 for all values of f1, . . . , fs; and
(6) to form a composite function CRS(fi, gj) defined by
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Abstract
A method and system for data modeling that incorporates the advantages of both traditional response surface methodology (RSM) and neural networks is disclosed. The invention partitions the parameters into a first set of s simple parameters, where observable data are expressible as low order polynomials, and c complex parameters that reflect more complicated variation of the observed data. Variation of the data with the simple parameters is modeled using polynomials; and variation of the data with the complex parameters at each vertex is analyzed using a neural network. Variations with the simple parameters and with the complex parameters are expressed using a first sequence of shape functions and a second sequence of neural network functions. The first and second sequences are multiplicatively combined to form a composite response surface, dependent upon the parameter values, that can be used to identify an accurate model.
60 Citations
9 Claims
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1. A method for constructing a composite response surface based on neural networks and selected functions, the method comprising providing a computer that is programmed:
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(1) to provide a set of h initial parameters that determine variation of provided data for a target variable, where each parameter corresponds to a coordinate in an h-dimensional parameter space G; (2) to decompose the h parameters into a first set of s simple parameters fi, numbered i=1, . . . , s, that may be used to describe the provided data with polynomials of total degree no greater than a selected number Ms, and a second set of c complex parameters gj, numbered j=1, . . . , c, that may be used to describe the provided data using neural networks, and with s+c=h, where s, c and Ms are selected positive integers; (3) to provide a simplex, having s+1 vertices, numbered k=1, . . . , s+1, and centered at a selected point in the space G; (4) to apply a neural network for each of the s+1 vertices, and to train each of the s+1 neural networks, using selected simulation data obtained by varying the parameters gj to generate a first sequence of network functions Rk(g1, . . . , gc); (5) to provide a second sequence of shape functions Pk(f1, . . . , fs) that satisfy the conditions Pk(f1, . . . , fs)=1 at the vertex numbered k and Pk(f1, . . . , fs)=0 at any vertex other than vertex number k, and Σ
Pk(f1, . . . , fs)=1 for all values of f1, . . . , fs; and(6) to form a composite function CRS(fi, gj) defined by - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
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Specification