A. N. Vasilyev, Ph. V. Porubayev, D. A. Tarkhov
Neural networking technique with models based on partial differential equations is applied to known incorrect problems which solution by routine approaches has difficulties. An approximate solution to the problem is found as output of neural network with some prescribed architecture. Network weights are determined in the process of stepwise network training grounded on some error functional minimization in general case. Given initial and boundary conditions for heat conduction equation allow us to solve the problem "forward", and such a problem is well known as a correct one. From the formal point of view these conditions allow us to search for the solution to the problem "backward", but it is an ill-posed problem now. Neural network approach gives us on opportunity to get a stable approximate solution – regularization of the problem. Another example of similar problem is an incorrect problem of temperature field evaluation according to approximately known point measurement data. Some regularization of this problem was realized by A.Samarsky and P.Vabishevich as a control problem via initial data reconstruction on the set of point data (boundary conditions are given). By the offered approach based on the neural network methodology both the solution of correct direct problem and regularization of incorrect inverse problem are constructed uniformly.
The function was used as a model solution of two-dimensional heat conduction equation that is to be recovered by numerical experiment. Values of the function were given approximately with an error in points of some discrete subset of the domain . Variants of , , value assignment are considered. Different variants of initial weights in neural network training process are analyzed. Two ways of training are considered: fixed neural networks and growing neural networks. Results of neurocomputing and some corresponding figures are given. Advantages of neural network approach and some possible generalizations are mentioned.