# Variational Bayes Deep Operator Network: A data-driven Bayesian solver for parametric differential equations

arXiv: https://arxiv.org/abs/2206.05655 · HTML: https://mlsanity.com/paper/2206.05655/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 67 references that are in the map.
- First posted on arXiv 12 Jun 2022.

## Builds on (most cited first, 14 of 14)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Challenges in Markov chain Monte Carlo for Bayesian neural networks](https://mlsanity.com/paper/1910.06539/) (2019)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs](https://mlsanity.com/paper/2111.02484/) (2021)
- [Fourier Neural Networks: A Comparative Study](https://mlsanity.com/paper/1902.03011/) (2019)
- [Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systems](https://mlsanity.com/paper/2201.13145/) (2022)
- [Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems](https://mlsanity.com/paper/2109.00538/) (2021)

## Built on by (most cited first, 11 of 11)

- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations](https://mlsanity.com/paper/2209.09750/) (2022)
- [Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media](https://mlsanity.com/paper/2405.19019/) (2024)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation](https://mlsanity.com/paper/2404.15731/) (2024)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [AI models of unstable flow exhibit hallucination](https://mlsanity.com/paper/2604.20372/) (2026)
- [PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference](https://mlsanity.com/paper/2607.05134/) (2026)

## Abstract (arXiv)

Neural network based data-driven operator learning schemes have shown tremendous potential in computational mechanics. DeepONet is one such neural network architecture which has gained widespread appreciation owing to its excellent prediction capabilities. Having said that, being set in a deterministic framework exposes DeepONet architecture to the risk of overfitting, poor generalization and in its unaltered form, it is incapable of quantifying the uncertainties associated with its predictions. We propose in this paper, a Variational Bayes DeepONet (VB-DeepONet) for operator learning, which can alleviate these limitations of DeepONet architecture to a great extent and give user additional information regarding the associated uncertainty at the prediction stage. The key idea behind neural networks set in Bayesian framework is that, the weights and bias of the neural network are treated as probability distributions instead of point estimates and, Bayesian inference is used to update their prior distribution. Now, to manage the computational cost associated with approximating the posterior distribution, the proposed VB-DeepONet uses textitvariational inference. Unlike Markov Chain Monte Carlo schemes, variational inference has the capacity to take into account high dimensional posterior distributions while keeping the associated computational cost low. Different examples covering mechanics problems like diffusion reaction, gravity pendulum, advection diffusion have been shown to illustrate the performance of the proposed VB-DeepONet and comparisons have also been drawn against DeepONet set in deterministic framework.