We develop a neural network approach to identifying parameters with steady-state solutions, locating such solutions, and determining their linear stability for systems of ordinary differential equations and dynamical systems with parameters. We first construct target functions that can be used to identify parameters with steady-state solution and the linear stability of such solutions. We then design a parameter-solution neural network (PSNN) that couples a parameter neural network and a solution neural network to approximate the target function, and develop efficient algorithms to train the PSNN and to locate steady-state solutions. We also present a theory of approximation of the target function by our PSNN. Numerical results are reported to show that our approach is robust in identifying the phase boundaries separating different regions in the parameter space corresponding to no solution or different numbers of solutions and in classifying the stability of solutions. These numerical results validate our analysis. Some potential improvements and future work are discussed. This is joint work with Yimeng Zhang, Alexander Cloninger, and Xiaochuan Tian.
