### Abstract

This chapter aims to draw the framework in which neural network (NN) solutions to the problem can be developed and studied; and to show how careful considerations on the fundamental issues naturally lead to the Wave-Net solution. The analysis not only attempts to justify the development of the Wave-Net but also refines its operational characteristics. The motivation for studying the functional estimation problem is the derivation of a modeling framework suitable for process control. The problem of deriving an estimate of an unknown function from empirical data is posed and studied in a theoretical level. Then, the problem is formulated in mathematical terms and the sources of the error related to any proposed solution to the estimation problem are identified. The chapter formulates basic algorithm whose new element is the pointwise presentation of the data and the dynamic evolution of the solution itself. The need for a multi-resolution framework in representing the unknown function is recognized and the wavelet transform is proposed as the essential vehicle to satisfy this requirement. The complete algorithm has been presented and identified as the modified Wave-Net model. Modeling examples demonstrate the properties of the derived solution.

Original language | English (US) |
---|---|

Pages (from-to) | 437-484 |

Number of pages | 48 |

Journal | Advances in Chemical Engineering |

Volume | 22 |

Issue number | C |

DOIs | |

State | Published - Jan 1 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Chemistry(all)
- Biomaterials
- Chemical Engineering(all)

### Cite this

*Advances in Chemical Engineering*,

*22*(C), 437-484. https://doi.org/10.1016/S0065-2377(08)60264-1

**Empirical Learning Through Neural Networks : The Wave-Net Solution.** / Koulouris, Alexandros; Bakshi, Bhavik R.; Stephanopoulos, George.

Research output: Contribution to journal › Article

*Advances in Chemical Engineering*, vol. 22, no. C, pp. 437-484. https://doi.org/10.1016/S0065-2377(08)60264-1

}

TY - JOUR

T1 - Empirical Learning Through Neural Networks

T2 - The Wave-Net Solution

AU - Koulouris, Alexandros

AU - Bakshi, Bhavik R.

AU - Stephanopoulos, George

PY - 1995/1/1

Y1 - 1995/1/1

N2 - This chapter aims to draw the framework in which neural network (NN) solutions to the problem can be developed and studied; and to show how careful considerations on the fundamental issues naturally lead to the Wave-Net solution. The analysis not only attempts to justify the development of the Wave-Net but also refines its operational characteristics. The motivation for studying the functional estimation problem is the derivation of a modeling framework suitable for process control. The problem of deriving an estimate of an unknown function from empirical data is posed and studied in a theoretical level. Then, the problem is formulated in mathematical terms and the sources of the error related to any proposed solution to the estimation problem are identified. The chapter formulates basic algorithm whose new element is the pointwise presentation of the data and the dynamic evolution of the solution itself. The need for a multi-resolution framework in representing the unknown function is recognized and the wavelet transform is proposed as the essential vehicle to satisfy this requirement. The complete algorithm has been presented and identified as the modified Wave-Net model. Modeling examples demonstrate the properties of the derived solution.

AB - This chapter aims to draw the framework in which neural network (NN) solutions to the problem can be developed and studied; and to show how careful considerations on the fundamental issues naturally lead to the Wave-Net solution. The analysis not only attempts to justify the development of the Wave-Net but also refines its operational characteristics. The motivation for studying the functional estimation problem is the derivation of a modeling framework suitable for process control. The problem of deriving an estimate of an unknown function from empirical data is posed and studied in a theoretical level. Then, the problem is formulated in mathematical terms and the sources of the error related to any proposed solution to the estimation problem are identified. The chapter formulates basic algorithm whose new element is the pointwise presentation of the data and the dynamic evolution of the solution itself. The need for a multi-resolution framework in representing the unknown function is recognized and the wavelet transform is proposed as the essential vehicle to satisfy this requirement. The complete algorithm has been presented and identified as the modified Wave-Net model. Modeling examples demonstrate the properties of the derived solution.

UR - http://www.scopus.com/inward/record.url?scp=0042482544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042482544&partnerID=8YFLogxK

U2 - 10.1016/S0065-2377(08)60264-1

DO - 10.1016/S0065-2377(08)60264-1

M3 - Article

AN - SCOPUS:0042482544

VL - 22

SP - 437

EP - 484

JO - Advances in Chemical Engineering

JF - Advances in Chemical Engineering

SN - 0065-2377

IS - C

ER -