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
HNS: An Efficient Hermite Neural Solver for Solving Time-Fractional Partial Differential Equations
AI总结：
针对基于L1插值的神经网络求解器精度受限问题，提出融合高阶Hermite插值显式格式与深度神经网络的HNS，在正、反问题及高维场景下精度更优，实现两类方法优势互补。
摘要：
神经网络求解器是利用深度学习技术求解时间分数阶偏微分方程的一种创新且极具前景的方法。L1插值逼近是神经网络求解器中处理时间分数阶导数的标准方法。但我们发现，基于L1插值逼近的神经网络求解器无法充分发挥神经网络的优势，且这类模型的精度受限于插值误差。本文提出了用于求解时间分数阶偏微分方程的高精度Hermite神经求解器（HNS）。具体而言，我们首先利用Hermite插值技术构建了分数阶导数的高阶显式逼近格式，并严格分析了其逼近精度。随后，结合深度神经网络的无限可微特性，我们将高阶Hermite插值显式逼近格式与深度神经网络相融合，提出了HNS。实验结果表明，无论是正问题还是反问题，以及在高维场景下，HNS的精度均高于基于L1格式的方法。这说明HNS相较于现有基于L1的方法，在精度和灵活性上均有显著提升，且突破了显式有限差分逼近方法通常仅局限于函数值插值的限制。因此，HNS并非数值计算方法与神经网络的简单结合，而是实现了二者优势的互补与相互强化。数据与代码可访问https://github.com/hsbhc/HNS获取。
英文摘要：
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 \url{https://github.com/hsbhc/HNS}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
