Yue Sun

Xuanle Zhao, Yue Sun, Tielin Zhang et al.

Neural operators improve conventional neural networks by expanding their capabilities of functional mappings between different function spaces to solve partial differential equations (PDEs). One of the most notable methods is the Fourier Neural Operator (FNO), which draws inspiration from Green's function method and directly approximates operator kernels in the frequency domain. However, after empirical observation followed by theoretical validation, we demonstrate that the FNO approximates kernels primarily in a relatively low-frequency domain. This suggests a limited capability in solving complex PDEs, particularly those characterized by rapid coefficient changes and oscillations in the solution space. Such cases are crucial in specific scenarios, like atmospheric convection and ocean circulation. To address this challenge, inspired by the translation equivariant of the convolution kernel, we propose a novel hierarchical Fourier neural operator along with convolution-residual layers and attention mechanisms to make them complementary in the frequency domain to solve complex PDEs. We perform experiments on forward and reverse problems of multiscale elliptic equations, Navier-Stokes equations, and other physical scenarios, and find that the proposed method achieves superior performance in these PDE benchmarks, especially for equations characterized by rapid coefficient variations.

Xuanle Zhao, Yue Sun, Tielin Zhang et al.

Spatiotemporal prediction plays an important role in solving natural problems and processing video frames, especially in weather forecasting and human action recognition. Recent advances attempt to incorporate prior physical knowledge into the deep learning framework to estimate the unknown governing partial differential equations (PDEs), which have shown promising results in spatiotemporal prediction tasks. However, previous approaches only restrict neural network architectures or loss functions to acquire physical or PDE features, which decreases the representative capacity of a neural network. Meanwhile, the updating process of the physical state cannot be effectively estimated. To solve the above mentioned problems, this paper proposes a physical-guided neural network, which utilizes the frequency-enhanced Fourier module and moment loss to strengthen the model's ability to estimate the spatiotemporal dynamics. Furthermore, we propose an adaptive second-order Runge-Kutta method with physical constraints to model the physical states more precisely. We evaluate our model on both spatiotemporal and video prediction tasks. The experimental results show that our model outperforms state-of-the-art methods and performs best in several datasets, with a much smaller parameter count.