Singularly-perturbed ordinary differential equations often exhibit Stokes' phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes curves. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a rational approximation based on a numerical leading-order solution using the adaptive Antoulas-Anderson (AAA) method. We show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. We explore the limitations of this approach, and show that for sufficiently small values of the asymptotic parameter, this approach breaks down; however, the range of validity may be extended by increasing the number of poles in the rational approximation. We finish by presenting a related nonlinear problem and discussing the challenges that arise when attempting to apply this method to nonlinear problems.

翻译：受奇异摄动的常微分方程通常表现出斯托克斯现象，即指数级微小振荡项在复平面上沿被称为斯托克斯曲线的路径出现和消失的现象。这类曲线起源于微分方程主导阶解的奇异点。在许多重要问题中，由于无法获得主导阶解的闭式表达式，因此定位这些奇异点极具挑战性。我们提出证据表明：利用自适应Antoulas-Anderson（AAA）方法，线性微分方程的解析主导阶解可替换为基于数值主导阶解的有理近似。研究证明，后续的指数渐近分析能准确预测解中存在的指数级微小行为。我们探讨了该方法的局限性——当渐近参数足够小时此方法会失效，但通过增加有理近似中的极点数量可扩展其有效范围。最后提出一个相关非线性问题，并讨论将此方法应用于非线性问题时面临的挑战。