  We develop a machine learning 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. Our approach begins with the construction of target functions
that can be used to identify parameters with steady-state solution and the
linear stability of such solutions. We 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 based on the neural network
kernel decomposition. 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 also validate our analysis. Although the primary focus in this study
centers on steady states of parameterized dynamical systems, our approach is
applicable generally to finding solutions for parameterized nonlinear systems
of algebraic equations. Some potential improvements and future work are
discussed.
