HANN: Homotopy auxiliary neural network for solving nonlinear algebraic equations

Solving nonlinear algebraic equations is a fundamental but challenging problem in scientific computations and also has many applications in system engineering. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing certain specific problems, there still exist certain weaknesses such as the initial value sensitivity, limited accuracy and slow convergence rate, particulary without flexible input for the neural network methods. In this paper, we propose a homotopy auxiliary neural network (HANN) for solving nonlinear algebraic equations which integrates the classical homotopy continuation method and popular physics-informed neural network. Consequently, the HANN-1 has strong learning ability and can rapidly give an acceptable solution for the problem which outperforms some known methods, while the HANN-2 can further improve its accuracy. Numerical results on the benchmark problems confirm that the HANN method can effectively solve the problems of determining the total number of solutions of a single equation, finding solutions of transcendental systems involving the absolute value function or trigonometric function, ill-conditioned and normal high-dimensional nonlinear systems and time-varying nonlinear problems, for which the Python's built-in Fsolve function exhibits significant limitations, even fails to work.