Deep Bayesian Operator Networks for forward and Inverse Problems

This research focuses on applying Bayesian Deep Operator Networks for practical parameter estimation in solving partial differential equations (PDEs). This innovative approach integrates deep learning with Bayesian inference, enabling the model to effectively capture unknown PDE parameter values and offer a probabilistic perspective on PDE solutions.

Our experiments utilize synthetic noisy datasets of the heat and reaction-diffusion equations. The promising results show remarkable alignment between the predicted and actual solutions. Crucially, the Bayesian framework supports the exploration of multimodal parameter distributions, revealing multiple plausible parameter states. This ability to recognize and quantify various potential solutions underpins the model’s power, offering insights into the probabilistic nature of the solutions and enhancing the robustness of predictions in real-world applications where the exact solution is unknown. These multimodal insights are pivotal in understanding the complex dynamics of the systems we study, presenting a comprehensive view of possible outcomes rather than a single predicted path.