In this paper, we introduce a high-precision Hermite neural network solver which employs Hermite interpolation technique to construct high-order explicit approximation schemes for fractional derivatives. By automatically satisfying the initial conditions, the construction process of the objective function is simplified, thereby reducing the complexity of the solution. Our neural networks are trained and fine-tuned to solve one-dimensional (1D) and two-dimensional (2D) time fractional Allen-Cahn equations with limited sampling points, yielding high-precision results. Additionally, we tackle the parameter inversion problem by accurately recovering model parameters from observed data, thereby validating the efficacy of the proposed algorithm. We compare the L2 relative error between the exact solution and the predicted solution to verify the robustness and accuracy of the algorithm. This analysis confirms the reliability of our method in capturing the fundamental dynamics of equations. Furthermore, we extend our analysis to three-dimensional (3D) cases, which is covered in the appendix, and provide a thorough evaluation of the performance of our method. This paper also conducts comprehensive analysis of the network structure. Numerical experiments indicate that the number of layers, the number of neurons in each layer, and the choice of learning rate play a crucial role in the performance of our solver.

中文翻译：

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基于 Hermite 神经求解器的时间分数阶 Allen-Cahn 方程数值模拟


在本文中，我们介绍了一种高精度 Hermite 神经网络求解器，它采用 Hermite 插值技术来构建分数阶导数的高阶显式逼近方案。通过自动满足初始条件，简化了目标函数的构造过程，从而降低了解的复杂度。我们的神经网络经过训练和微调，可以求解采样点有限的一维 （1D） 和二维 （2D） 时间分数 Allen-Cahn 方程，从而产生高精度的结果。此外，我们通过从观测数据中准确恢复模型参数来解决参数反演问题，从而验证了所提算法的有效性。我们比较了精确解和预测解之间的 L2 相对误差，以验证算法的稳健性和准确性。该分析证实了我们的方法在捕获方程的基本动力学方面的可靠性。此外，我们将分析扩展到附录中介绍的三维 （3D） 案例，并对我们方法的性能进行了全面评估。本文还对网络结构进行了综合分析。数值实验表明，层数、每层中的神经元数以及学习率的选择对求解器的性能起着至关重要的作用。