In most cases, differential equations in the real world don't have their solutions as closed forms. So numerical methods are prefered to obtain an approximated solution. However, the most popular numerical method to solve differential equations, FEM (finite element method) has some drawbacks with their solutions. It provides its solution as a set of discrete values at the specific points in the domain. Therefore, the solution is not differentiable and it is not straitforward to obtain the values of the solution at the other points in the domain. Moreover, it requires lots of memory in order to present the solution at the specific points in the domain. Neural networks are good alternative tools for overcoming these drawbacks of FEM, because they are universal approximators, have continuous and differentiable outputs and require small memory space. Up to now, the MLP-based and the RBF-based techniques are proposed respectively as differential equation solvers using neural networks. The MLP-based technieque among them obtains the solution directly, that is from the MLP itself by approximating the solution with the MLP. In this thesis, The MLP-based technique that approximates not the solution but the derivative of the solution with the MLP is proposed. This technique obtains the solution indirectly, that is from the integrated function of the MLP with respect to the input. It can be a issue how to deal with the integration constants in the training phase of the MLP. This thesis proposed an algorithm for determining the integration constants using the pseudo-inverse. The proposed structure with the algorithm has been tested on several ODEs and PDEs with and without the noise. The results show that the proposed indirect method is more robust against the overfitting problems than the conventional direct one especially on the ODEs without the noise. And it requires less memory spaces than the RBF-based techniques. The results from the simulation with the noise show that the MLP-based techniques yields better solutions than the RBF-based techniques bacause of their small number of parameters and the abilities of choosing their stopping criteria to avoid learning the noise.
