R-adaptive DeepONet: Learning Solution Operators for PDEs with Discontinuous Solutions Using an R-adaptive Strategy

Abstract: DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh (R-adaptive) method, we propose an R-adaptive DeepONet method, which contains the following components: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the (linear) interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the final error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Therefore, the R-adaptive DeepONet overcomes the limitations of DeepONet, and can reduce the approximation error for problems with discontinuous solutions. Numerical experiments on PDEs with discontinuous solutions, including the linear advection equation, the Burgers' equation with low viscosity, and the compressible Euler equations of gas dynamics, are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.