Numerically solving partial differential equations 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 partial differential equations 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 a global receptive field. This mechanism enhances the model's comprehension of the input's context and enables data-dependent weight for different partial differential equations instances. To measure how effective the layers are in solving partial differential equations, we conduct experiments on Diffusion-Reaction equation and Navier Stokes equation. Our findings indicate Hyena Neural operator can serve as an efficient and accurate model for learning partial differential equations solution operator. The data and code used can be found at: https://github.com/Saupatil07/Hyena-Neural-Operator

翻译：数值求解偏微分方程通常需要精细的离散化以解析必要的时空尺度，这可能导致高昂的计算成本。深度学习的最新进展提供了一种求解偏微分方程的新方法，即使用神经算子。神经算子是一种神经网络架构，能够学习函数空间之间的映射，并具有基于数据求解偏微分方程的能力。本研究采用了一种名为Hyena的新型神经算子，该算子使用由多层感知机参数化的长卷积滤波器。Hyena算子是一种具有次二次复杂度的运算，并利用状态空间模型参数化具有全局感受野的长卷积。这种机制增强了模型对输入上下文的理解，并使得针对不同的偏微分方程实例能够实现数据依赖的权重。为衡量各层在求解偏微分方程中的有效性，我们在扩散-反应方程和纳维-斯托克斯方程上进行了实验。研究结果表明，Hyena神经算子可作为学习偏微分方程解算子的高效且精确模型。所用数据和代码可访问：https://github.com/Saupatil07/Hyena-Neural-Operator