Numerically solving partial differential equations (PDEs) typically requires
fine discretization to resolve necessary spatiotemporal scales, which can be
computationally expensive. Recent advances in deep learning have provided a new
approach to solving PDEs that involves the use of neural operators. Neural
operators are neural network architectures that learn mappings between function
spaces and have the capability to solve partial differential equations based on
data. This study utilizes a novel neural operator called Hyena, which employs a
long convolutional filter that is parameterized by a multilayer perceptron. The
Hyena operator is an operation that enjoys sub-quadratic complexity and state
space model to parameterize long convolution that enjoys global receptive
field. This mechanism enhances the model's comprehension of the input's context
and enables data-dependent weight for different PDE instances. To measure how
effective the layers are in solving PDEs, we conduct experiments on Burger's
equation and Navier Stokes equation. Our findings indicate Hyena Neural
operator can serve as an efficient and accurate model for learning PDEs'
solution operator. The data and code used can be found at:
this https URL

该研究利用一种新型神经算子 Hyena，通过深度学习解决偏微分方程问题，实验结果表明，该算子能够有效地学习和预测偏微分方程的解。