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.

翻译：我们开发了一种数据驱动的机器学习方法，用于识别具有稳态解的参数、定位此类解，并确定常微分方程和参数化动力系统解的线性稳定性。我们的方法首先为这些任务构建目标函数，然后设计一个参数-解耦神经网络（PSNN），该网络耦合参数神经网络和解神经网络以逼近目标函数。我们进一步开发了高效算法来训练PSNN并定位稳态解。基于核分解，我们建立了PSNN逼近目标函数的近似理论。数值结果表明，我们的方法在寻找解、识别相边界以及对参数区域内解的稳定性进行分类方面具有鲁棒性。这些数值结果也验证了我们的理论分析。虽然本研究聚焦于参数化动力系统的稳态，但我们的方法不依赖于具体方程形式，可普遍应用于寻找参数化非线性代数方程组的解。文中还讨论了一些潜在的改进方向和未来工作。