Fast Solving Partial Differential Equations via Imitative Fourier Neural Operator

Neural operators are a class of neural networks to learn mappings between infinite-dimensional function spaces, and recent studies have shown that using neural operators to solve partial differential equations is a promising direction. However, the latest neural operator-based algorithms still leave space for increasing the running speed of algorithms while maintaining accuracy. In this work, we propose a new neural operator by using a neural network to "imitate" the process of the Fourier Transform to solve partial differential equations, called the Imitative Fourier Neural Operator (IFNO). The advantage of this method is to use the property of Fourier Transform to transform the partial differential equation into an algebraic equation, which reduces the fitting difficulty of the neural network and improves the training speed. We compare the proposed algorithm with several state-of-the-art designs on different benchmark instances. The experimental results confirm the effectiveness and performance of the proposed method for solving partial differential equations.