Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for addressing time-fractional derivatives within neural network solvers. However, we have discovered that neural network solvers based on L1 interpolation approximation are unable to fully exploit the benefits of neural networks, and the accuracy of these models is constrained to interpolation errors. In this paper, we present the high-precision Hermite Neural Solver (HNS) for solving time-fractional partial differential equations. Specifically, we first construct a high-order explicit approximation scheme for fractional derivatives using Hermite interpolation techniques, and rigorously analyze its approximation accuracy. Afterward, taking into account the infinitely differentiable properties of deep neural networks, we integrate the high-order Hermite interpolation explicit approximation scheme with deep neural networks to propose the HNS. The experimental results show that HNS achieves higher accuracy than methods based on the L1 scheme for both forward and inverse problems, as well as in high-dimensional scenarios. This indicates that HNS has significantly improved accuracy and flexibility compared to existing L1-based methods, and has overcome the limitations of explicit finite difference approximation methods that are often constrained to function value interpolation. As a result, the HNS is not a simple combination of numerical computing methods and neural networks, but rather achieves a complementary and mutually reinforcing advantages of both approaches. The data and code can be found at .

中文翻译：

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HNS：一种用于求解时间分数阶偏微分方程的高效 Hermite 神经求解器

神经网络求解器代表了一种利用深度学习技术解决时间分数阶偏微分方程的创新且有前途的方法。L1 插值近似是神经网络求解器中处理时间分数导数的标准方法。然而，我们发现基于L1插值近似的神经网络求解器无法充分利用神经网络的优势，并且这些模型的精度受限于插值误差。在本文中，我们提出了用于求解时间分数阶偏微分方程的高精度埃尔米特神经求解器（HNS）。具体来说，我们首先使用Hermite插值技术构建了分数阶导数的高阶显式逼近方案，并严格分析了其逼近精度。随后，考虑到深度神经网络的无限可微特性，我们将高阶 Hermite 插值显式逼近方案与深度神经网络相结合，提出了 HNS。实验结果表明，对于正向和逆向问题以及高维场景，HNS 比基于 L1 方案的方法取得了更高的精度。这表明，与现有的基于 L1 的方法相比，HNS 显着提高了精度和灵活性，并且克服了显式有限差分逼近方法通常受限于函数值插值的局限性。因此，HNS并不是数值计算方法和神经网络的简单结合，而是实现了两种方法的优势互补、相互促进。数据和代码可以在 找到。