Deep Operator Networks for Bayesian Parameter Estimation in PDEs
This provides a computationally efficient method for uncertainty quantification in PDE surrogate modeling, which is incremental as it integrates existing techniques.
The paper tackles the problem of solving partial differential equations (PDEs) and estimating unknown parameters by combining Deep Operator Networks with Physics-Informed Neural Networks, achieving robust and accurate solutions with uncertainty quantification in noisy or incomplete scenarios.
We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven learning with physical constraints, our method achieves robust and accurate solutions across diverse scenarios. Bayesian training is implemented through variational inference, allowing for comprehensive uncertainty quantification for both aleatoric and epistemic uncertainties. This ensures reliable predictions and parameter estimates even in noisy conditions or when some of the physical equations governing the problem are missing. The framework demonstrates its efficacy in solving forward and inverse problems, including the 1D unsteady heat equation and 2D reaction-diffusion equations, as well as regression tasks with sparse, noisy observations. This approach provides a computationally efficient and generalizable method for addressing uncertainty quantification in PDE surrogate modeling.