# Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification

arXiv: https://arxiv.org/abs/2408.00681 · HTML: https://mlsanity.com/paper/2408.00681/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 56 references that are in the map.
- First posted on arXiv 1 Aug 2024.

## Builds on (most cited first, 30 of 31)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [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)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [Expressive Priors in Bayesian Neural Networks: Kernel Combinations and Periodic Functions](https://mlsanity.com/paper/1905.06076/) (2019)
- [Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs](https://mlsanity.com/paper/2111.02484/) (2021)
- [Doubly Robust Bayesian Inference for Non-Stationary Streaming Data with β-Divergences](https://mlsanity.com/paper/1806.02261/) (2018)
- [IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning](https://mlsanity.com/paper/2302.03271/) (2023)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [A multi-fidelity deep operator network (DeepONet) for fusing simulation and monitoring data: Application to real-time settlement prediction during tunnel construction](https://mlsanity.com/paper/2310.00057/) (2023)
- [Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet](https://mlsanity.com/paper/2104.01996/) (2021)

## Built on by (most cited first, 4 of 4)

- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning](https://mlsanity.com/paper/2603.16757/) (2026)
- [PGD-NO: A Neural Operator with Precomputed Geometry Decomposition for 3D Million-scale Physics Simulations](https://mlsanity.com/paper/2607.08025/) (2026)
- [Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields](https://mlsanity.com/paper/2608.09071/) (2026)

## Abstract (arXiv)

We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R'enyi's α-divergence to learn complex operators while quantifying uncertainty. By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. The use of R'enyi's α-divergence, instead of the Kullback-Leibler divergence (KLD), commonly used in standard variational inference, mitigates issues related to prior misspecification that are prevalent in Variational Bayesian DeepONets. This approach offers enhanced flexibility and robustness. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error and improving the negative log-likelihood on the test set. Our framework's efficacy is validated across various mechanical systems, where it outperforms both deterministic and standard KLD-based VI DeepONets in predictive accuracy and uncertainty quantification. The hyperparameter α, which controls the degree of robustness, can be tuned to optimise performance for specific problems. We apply this approach to a range of mechanics problems, including gravity pendulum, advection-diffusion, and diffusion-reaction systems. Our findings underscore the potential of α-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.