# Integration of Functional Link Neural Networks into a Parameter Estimation Methodology
###### Abstract
The basic building blocks of every ANN are artificial neurons, i.e., simple mathematical models (functions). Typical ANNs comprise thousands or millions of artificial neurons (i.e., nonlinear processing units) connected via (synaptic) weights. ANNs can \"learn\" a task by adjusting these weights. Neurons receive inputs with their associated weights, transform those inputs using activation functions, and pass the transformed information as outputs. It has been theoretically proved that ANNs can approximate any continuous mapping to arbitrary precision without any assumptions [19; 20; 21; 22]. Furthermore, without any knowledge of underlying principles, ANNs can determine unknown interactions between the input and output performances of a process because of their data-driven and self-adaptive properties. Accordingly, the functional correlation between the input and output quality characteristics in RD can be modeled and analyzed by NNs without any assumptions. The integration of an NN into the experiment design procedure of an RD model has been mentioned in [PERSON] et al. [23] and [PERSON] et al. [24]. In recent times, [PERSON] and [PERSON] [25] presented a feed-forward NN-based RSM that improved the precision of estimations without additional experiments. [PERSON] et al. [26] proposed an NN-based estimation method that identified a new screening procedure to determine the optimum transfer function, so that a more accurate solution can be obtained. A genetic algorithm with NNs has been executed in [PERSON] and [PERSON] [27], [PERSON] et al. [28], [PERSON] et al. [29], [PERSON] [30], [PERSON] and [PERSON] [31], [PERSON] et al. [32], and [PERSON] et al. [33] as an estimation method to investigate the optimal quality characteristics with associated control factor settings in the RD model without the use of estimation formulas. [PERSON] et al. [34] introduced an efficient optimization method by combining the RSM and a genetic algorithm (GA) to find the optimal topology of ANNs for predicting color changes in rehydrated apple cubes. Therefore, the main objective is to propose a new dual-response estimation approach based on NNs. First, the normal quadratic process mean and standard deviation functions in RD are estimated using the proposed functional-link-NN-based estimation method. Second, the Bayesian information criterion (BIC) is used to quantify the magnitude of neurons in the hidden layer, which affects the coefficients in the estimated input-output equations. Finally, the results of the case study are presented to verify the effectiveness of the proposed NN-based estimation method compared with the LSM-based RSM. The graphical overview of the proposed NN-based estimation method is demonstrated in Figure 1.  
The remainder of the study is organized as follows: Section 2 introduces the conventional LSM-based RSM. Section 3 describes the functional-link-NN-based dual-response estimation model. Section 4 explains the outputs generated by the proposed NN-based estimation method and analyzes the LSM-based RSM based on the results of the case study. Finally, Section 5 concludes the study and describes further studies.