What makes this particular use of neural networks so attractive in many applications is that they express the ability to learn, though this remarkable property might be challenged by some researchers. In terms of ANNs, “learning” simply means changing the weights and biases of the network in response to some input data. Once a particular learning algorithm succeeds, programming the network to have a particular unequivocal performance is not vitally important. In other words, we need no prior knowledge to adjust the weights and biases. The designed neural architecture adjusts its parameters for a learning algorithm. That is, the error alters as the weights and bias term are changed. In this sense, the neural network learns from experience. Therefore, the back-propagation algorithm is the most widely-used method for feed-forward networks. The learning rules are mathematical formalizations that are believed to be more effective in the ANN’s performance. To fine-tune the neural network, the network parameters are first quantified with arbitrary initial guesses; then, the neural network calculates the output for each input signal. Next, the defined error rule is employed by substituting the proposed network model instead of the solution function in the origin problem. To train the present network, we have employed an optimization technique that in turn required the computation of the gradient of the error with respect to the net parameters.

Now, a suitable error correction rule must be initially used for single units training. This rule essentially drives the output error of the network to zero. We start with the classical generalized delta learning rule and give a brief description for its performance. Throughout this section, an attempt is made to point out the criterion function that is minimized by using this rule. Learning in neural nets is appropriate selecting the connection weights, which yields to minimize the error function on a set of mesh points. During the training, the initial parameters *a*_{i,j} are put into the network and flow through the network generating a real value on the output unit. As seen in the last part, the calculated output is compared with the desired one, and an error is computed. The differentiable cost functions *EW* and *EH* are always decreasing in the opposite direction of its derivative. It means that if we want to find one of the local minima of this function starting from a initial guess. We employ the supervised back-propagation learning algorithm to reach this goal. The mentioned self learning mechanism starts with randomly quantifying the initial parameters *a*_{i,j} (for *i* = 0, …, *n*; *j* = 0, …, *m*). The mentioned algorithm is well presented for wave-like equation as follows:

$$\begin{array}{}{\displaystyle {a}_{i,j}(r+1)={a}_{i,j}(r)+\mathrm{\Delta}{a}_{i,j}(r),}\end{array}$$(15)

$$\begin{array}{}{\displaystyle \mathrm{\Delta}{a}_{i,j}(r)=-\eta .\frac{\mathrm{\partial}E{W}_{p,q}}{\mathrm{\partial}{a}_{i,j}}+\gamma .\mathrm{\Delta}{a}_{i,j}(r-1),}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}p=0,...,{n}^{\prime};q=0,...,{m}^{\prime},\end{array}$$

where *η* and *γ* are the learning rate and momentum term, respectively. In the above, the index *r* in *a*_{i,j}(*r*) ascribes to the repetition number and the subscript *i*, *j* in *a*_{i,j} is the label of the training connection weight. Moreover, *a*_{i,j}(*r* + 1) and *a*_{i,j}(*r*) depict the adjusted and current weight parameter, respectively. To complete the derivation of learning procedure for the output layer weights, the above partial derivative can be expressed as follows:

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}E{W}_{p,q}}{\mathrm{\partial}{a}_{i,j}}}& =({\stackrel{~}{u}}_{tt}({x}_{p},{t}_{q})-k({x}_{p},{t}_{q}){\stackrel{~}{u}}_{xx}({x}_{p},{t}_{q})-h({x}_{p},{t}_{q}))\\ & \times \left(\frac{\mathrm{\partial}{\stackrel{~}{u}}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}-k({x}_{p},{t}_{q})\frac{\mathrm{\partial}{\stackrel{~}{u}}_{xx}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}\right),\end{array}$$

where

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\stackrel{~}{u}}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=\frac{\mathrm{\partial}{A}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+x(1-x)\left(2\frac{\mathrm{\partial}{N}_{t}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}+t\frac{\mathrm{\partial}{N}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}\right),\end{array}$$

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\stackrel{~}{u}}_{xx}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=\frac{\mathrm{\partial}{A}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+{t}_{q}\left(-2\left(\frac{\mathrm{\partial}N({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}-\frac{\mathrm{\partial}N({x}_{p},0)}{\mathrm{\partial}{a}_{i,j}}\right)\right.\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+(2-4{x}_{p})\left(\frac{\mathrm{\partial}{N}_{x}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}-\frac{\mathrm{\partial}{N}_{x}({x}_{p},0)}{\mathrm{\partial}{a}_{i,j}}\right)\\ \left.\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+({x}_{p}-{x}_{p}^{2})\left(\frac{\mathrm{\partial}{N}_{xx}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}\right)-\frac{\mathrm{\partial}{N}_{xx}({x}_{p},0)}{\mathrm{\partial}{a}_{i,j}}\right),\end{array}$$

and

$$\begin{array}{}\phantom{\rule{2em}{0ex}}\frac{\mathrm{\partial}N({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}={x}_{p}^{i}{t}_{q}^{j},\frac{\mathrm{\partial}{N}_{x}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=i{x}_{p}^{i-1}{t}_{q}^{j},\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial}{N}_{xx}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=i(i-1){x}_{p}^{i-2}{t}_{q}^{j},\\ \frac{\mathrm{\partial}{N}_{t}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=j{x}_{p}^{i}{t}_{q}^{j-1},\frac{\mathrm{\partial}{N}_{tt}({x}_{p},{t}_{q})}{\mathrm{\partial}{a}_{i,j}}=j(j-1){x}_{p}^{i}{t}_{q}^{j-2}.\end{array}$$

The above computational process can similarly be employed for heat-like equation. To prevent taking much of the time on this part, the behavior of adjusting weight parameters for this equation is not provided. It should be mentioned clearly that Matlab *v*7.10 is a high quality and easy to use mathematical computing software, which researchers and students can employ to omit wasting time and enhance the accuracy of calculations.

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