The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

中文翻译：

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求解普通和时间分数 Black-Scholes 方程的新 ANN 方法

本研究的主要目的是引入一个 2 层人工神经网络 (ANN)，用于求解分数阶或普通阶的 Black-Scholes 偏微分方程 (PDE)。首先，采用离散化方法将模型变为常微分方程（ODE）序列。随后，这些 ODE 中的每一个都在 ANN 的帮助下求解。Adam 优化被用作学习范式，因为它可以在接近实际最优解时添加减慢优化过程的预知。该模型还利用微调来加速过程和域映射以应对无限域问题。最后，报告了求解几种 Black-Scholes 模型的方法的准确性、速度和收敛性。