A recursive Bayesian neural network for constitutive modeling of sands under monotonic loading

In geotechnical engineering, constitutive models play a crucial role in describing soil behavior under varying loading conditions. Data-driven deep learning (DL) models offer a promising alternative for developing predictive constitutive models. When prediction is the primary focus, quantifying the predictive uncertainty of a trained DL model and communicating this uncertainty to end users is crucial for informed decision-making. This study proposes a recursive Bayesian neural network (rBNN) framework, which builds upon recursive feedforward neural networks (rFFNNs) by introducing generalized Bayesian inference for uncertainty quantification. A significant contribution of this work is the incorporation of a sliding window approach in rFFNNs, allowing the models to effectively capture temporal dependencies across load steps. The rBNN extends this framework by treating model parameters as random variables, with their posterior distributions inferred using generalized variational inference. The proposed framework is validated on two datasets: (i) a numerically simulated consolidated drained (CD) triaxial dataset employing a hardening soil model and (ii) an experimental dataset comprising 28 CD triaxial tests on Baskarp sand. Comparative analyses with LSTM, Bi-LSTM, and GRU models demonstrate that the deterministic rFFNN achieves superior predictive accuracy, attributed to its transparent structure and sliding window design. While the rBNN marginally trails in accuracy for the experimental case, it provides robust confidence intervals, addressing data sparsity and measurement noise in experimental conditions. The study underscores the trade-offs between deterministic and probabilistic approaches and the potential of rBNNs for uncertainty-aware constitutive modeling.

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

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.