Neural network applications to the reconstruction of Coulomb field in potential flows

We apply an artificial neural network (ANN) to solve the Cauchy problem by modeling the flow of heavy impurities in potential flows of an incompressible fluid generated by a harmonic potential. We construct an ANN that does not require training to represent the gradient of the Coulomb potential, demonstrating high approximation accuracy. We establish the connection between iterative methods for solving the Cauchy problem and ANNs. Specifically, we design an ANN that implements the Runge-Kutta method, including a fourth-order accurate version, where weight coefficients are predefined at the design stage, eliminating the need for training. We perform a test calculation to compare impurity dynamics simulated using the developed ANN with results from a software package for solving the Cauchy problem. We analyze different initial data configurations and visualize the simulated flow of heavy impurities. Additionally, we demonstrate the practical application of the ANN for locating sources of the Coulomb potential.