Finding multiple solutions of odes with neural networks

Applications of neural networks to numerical problems have gained increasing interest. Among different tasks, finding solutions of ordinary differential equations (ODEs) is one of the most intriguing problems, as there may be advantages over well established and robust classical approaches. In this pa- per, we propose an algorithm to find all possible solutions of an ordinary differential equation that has multiple solutions, using artificial neural networks. The key idea is the introduction of a new loss term that we call the interaction loss. The interaction loss prevents different solutions families from coinciding. We carried out experiments with two nonlinear differential equations, all admitting more than one solution, to show the effectiveness of our algorithm, and we performed a sensitivity analysis to investigate the impact of different hyper-parameters.