Solving nonlinear algebraic equations is a fundamental yet challenging problem in scientific computations and real-world applications. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing some specific problems, there still exist certain weaknesses such as initial value sensitivity, limited accuracy and slow convergence rate. In this paper, we propose a homotopy auxiliary neural network for solving nonlinear algebraic equations which integrates the classical homotopy continuation method and the popular physics-informed neural networks. Consequently, the HANN has strong learning ability and can rapidly give an acceptable solution which outperforms some known methods,while iterative using the HANN can further improve its accuracy. Numerical results on the benchmark examples confirm that the HANN 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, for which the Python’s built-in Fsolve function exhibits significant limitations, even fails to work. Finally, we successfully apply the HANN to solve the time-varying nonlinear equations and the inverse kinematics problem of a 6-degree-of-freedom robot manipulator.