# Bayesian Inversion with Neural Operator (BINO) for Modeling Subdiffusion: Forward and Inverse Problems

arXiv: https://arxiv.org/abs/2211.11981 · HTML: https://mlsanity.com/paper/2211.11981/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 43 references that are in the map.
- First posted on arXiv 22 Nov 2022.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)

## Built on by (most cited first, 1 of 1)

- [Laplace-fPINNs: Laplace-based fractional physics-informed neural networks for solving forward and inverse problems of subdiffusion](https://mlsanity.com/paper/2304.00909/) (2023)

## Abstract (arXiv)

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.