Transfer learning of neural operators for partial differential equations based on sparse network λ-FNO

When the solution domain, internal parameters, and initial and boundary conditions of partial differential equation (PDE) are changed, many potential characteristics of the equation's solutions are still similar. This provides the possibility to reduce the cost of PDE operator learning through transfer learning methods. Based on Fourier neural operator (FNO), we propose a novel sparse neural operator network named λ-FNO. By introducing the λ parameter matrix and using a new pruning method to make the network sparse, the operator learning ability of λ-FNO is greatly improved. Using λ-FNO can efficiently learn the operator from the discrete initial function space on the uniform grid to the discrete equation's solution space on the unstructured grid, which is not available in FNO. Finally, we apply λ-FNO to several specific transfer tasks of partial differential equations under conditional distributions to demonstrate its excellent transferability. The experimental results show that when the shape of the solution domain of the equation or its internal parameters change, our framework can capture the potential invariant information of its solution and complete related transfer learning tasks with less cost, higher accuracy, and faster speed. In addition, the sparse framework has excellent extension and can be easily extended to other network architectures to enhance its performance. Our model and data generation code can get through https://github.com/Xumouren12/TL-FNO.