Bayesian Inversion with Neural Operator (BINO) for modeling subdiffusion: Forward and inverse problems

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 modeling 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.