Deep Operator Networks for Bayesian Parameter Estimation in PDEs

TL;DR

This paper introduces a combined Deep Operator Network and Physics-Informed Neural Network framework for solving PDEs and estimating parameters with Bayesian uncertainty quantification, demonstrating robustness and efficiency across various problems.

Contribution

It presents a novel integration of DeepONets with PINNs and Bayesian inference for PDE parameter estimation, enhancing robustness and uncertainty quantification.

Findings

Accurately solves forward and inverse PDE problems.

Provides comprehensive uncertainty quantification.

Demonstrates effectiveness on noisy and sparse data.

Abstract

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.