The quasipotential function allows for comprehension and prediction of the escape mechanisms from metastable states in nonlinear dynamical systems. This function acts as a natural extension of the potential function for non-gradient systems and it unveils important properties such as the maximum likelihood transition paths, transition rates and expected exit times of the system. Here, we leverage on machine learning via the combination of two data-driven techniques, namely a neural network and a sparse regression algorithm, to obtain symbolic expressions of quasipotential functions. The key idea is first to determine an orthogonal decomposition of the vector field that governs the underlying dynamics using neural networks, then to interpret symbolically the downhill and circulatory components of the decomposition. These functions are regressed simultaneously with the addition of mathematical constraints. We show that our approach discovers a parsimonious quasipotential equation for an archetypal model with a known exact quasipotential and for the dynamics of a nanomechanical resonator. The analytical forms deliver direct access to the stability of the metastable states and predict rare events with significant computational advantages. Our data-driven approach is of interest for a wide range of applications in which to assess the fluctuating dynamics.

翻译：准势函数有助于理解和预测非线性动力系统中亚稳态的逃逸机制。该函数作为非梯度系统势函数的自然延伸，揭示了系统的关键特性，如最大似然跃迁路径、跃迁速率及预期逃出时间。本文通过结合两种数据驱动技术——神经网络与稀疏回归算法——利用机器学习方法获取准势函数的符号表达式。核心思路是：首先使用神经网络确定支配底层动力学向量场的正交分解，随后对分解的下坡分量与环流分量进行符号化解析。这些函数在添加数学约束的条件下被同步回归。我们证明，该方法能为具有已知精确准势的典型模型及纳米机械谐振器的动力学特性，发现简约的准势方程。其解析形式可直接揭示亚稳态的稳定性，并以显著的计算优势预测罕见事件。我们的数据驱动方法在评估涨落动力学的广泛应用中具有重要意义。