  Fractional diffusion equations have been an effective tool for modeling
anomalous diffusion in complicated systems. However, traditional numerical
methods require expensive computation cost and storage resources because of the
memory effect brought by the convolution integral of time fractional
derivative. We propose a Bayesian Inversion with Neural Operator (BINO) to
overcome the difficulty in traditional methods as follows. We employ a deep
operator network to learn the solution operators for the fractional diffusion
equations, allowing us to swiftly and precisely solve a forward problem for
given inputs (including fractional order, diffusion coefficient, source terms,
etc.). In addition, we integrate the deep operator network with a Bayesian
inversion method for modelling a problem by subdiffusion process and solving
inverse subdiffusion problems, which reduces the time costs (without suffering
from overwhelm storage resources) significantly. A large number of numerical
experiments demonstrate that the operator learning method proposed in this work
can efficiently solve the forward problems and Bayesian inverse problems of the
subdiffusion equation.
