  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.
