HNS: An Efficient Hermite Neural Solver for Solving Time-Fractional
  Partial Differential Equations
Jie Hou Zhiying Ma Shihui Ying Ying Li
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}.