🤖 AI Summary
Traditional variational Bayesian DeepONets suffer from inaccurate uncertainty quantification due to misspecified priors. To address this, we propose α-DeepONet, a Bayesian DeepONet framework based on Rényi α-divergence. Our method is the first to incorporate generalized variational inference into the DeepONet architecture, replacing the standard KL divergence with the more flexible α-divergence for posterior approximation—thereby significantly improving prior robustness. We further introduce a dynamic, self-adaptive mechanism for the hyperparameter α, enhancing generalization across diverse physical systems. Integrating Bayesian neural networks with the DeepONet’s branch-trunk architecture, α-DeepONet is evaluated on canonical mechanical systems—including inverted pendulum, convection-diffusion, and diffusion-reaction problems. Results demonstrate superior predictive accuracy and uncertainty calibration compared to both deterministic DeepONet and KL-based variational variants. This work establishes a more reliable and adaptive Bayesian paradigm for complex operator learning.
📝 Abstract
We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R'enyi's $alpha$-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 $alpha$-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 $alpha$, 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 $alpha$-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.