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

We develop a data-driven 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 first constructs target functions for these tasks, then designs a parameter-solution neural network (PSNN) that couples a parameter neural network and a solution neural network to approximate the target functions. We further develop efficient algorithms to train the PSNN and locate steady-state solutions. An approximation theory for the target functions with PSNN is developed based on kernel decomposition. Numerical results are reported to show that our approach is robust in finding solutions, identifying phase boundaries, and classifying solution stability across parameter regions. These numerical results also validate our analysis. While this study focuses on steady states of parameterized dynamical systems, our approach is equation-free and is applicable generally to finding solutions for parameterized nonlinear systems of algebraic equations. Some potential improvements and future work are discussed.