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.

翻译：我们发展了一种机器学习方法，用于识别具有稳态解的参数、定位此类解，并确定常微分方程组及含参数动力系统的线性稳定性。该方法首先构建目标函数，该函数可用于识别具有稳态解的参数及其线性稳定性。我们设计了一种参数-解神经网络（PSNN），该网络通过耦合参数神经网络与解神经网络来逼近目标函数，并开发了高效算法用于训练PSNN及定位稳态解。同时，我们基于神经网络核分解提出了PSNN逼近目标函数的理论。数值结果表明，该方法在识别参数空间中对应无解或不同数量解的不同区域的相边界，以及分类解的稳定性方面具有稳健性。这些数值结果也验证了我们的理论分析。尽管本研究主要关注参数化动力系统的稳态，但该方法普遍适用于求解参数化非线性代数方程组。最后讨论了潜在的改进方向与未来工作。