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We develop a machine learning approach to identifying parameters withsteady-state solutions, locating such solutions, and determining their linearstability for systems of ordinary differential equations and dynamical systemswith parameters. Our approach begins with the construction of target functionsthat can be used to identify parameters with steady-state solution and thelinear stability of such solutions. We design a parameter-solution neuralnetwork (PSNN) that couples a parameter neural network and a solution neuralnetwork to approximate the target function, and develop efficient algorithms totrain the PSNN and to locate steady-state solutions. We also present a theoryof approximation of the target function by our PSNN based on the neural networkkernel decomposition. Numerical results are reported to show that our approachis robust in identifying the phase boundaries separating different regions inthe parameter space corresponding to no solution or different numbers ofsolutions and in classifying the stability of solutions. These numericalresults also validate our analysis. Although the primary focus in this studycenters on steady states of parameterized dynamical systems, our approach isapplicable generally to finding solutions for parameterized nonlinear systemsof algebraic equations. Some potential improvements and future work arediscussed.

A neural network kernel decomposition for learning multiple steady states in parameterized dynamical systems