Deep Operator Networks for Bayesian Parameter Estimation in PDEs

2025-01-18

AI Summary

Parameter estimation for partial differential equations (PDEs) with unknown parameters remains challenging under data sparsity, measurement noise, or incomplete physical models. Method: This paper proposes a Bayesian variational inference framework that synergistically integrates Deep Operator Networks (DeepONets) with physics-informed neural networks (PINNs). It is the first to embed DeepONets within variational inference, enabling unified quantification of aleatoric and epistemic uncertainties while jointly optimizing forward simulation and inverse parameter estimation—without requiring exact prior knowledge of the governing PDEs. Contribution/Results: The framework achieves robust parameter inference and calibrated uncertainty quantification even with imperfect physical models. Evaluated on benchmark problems—including 1D unsteady heat conduction and 2D reaction–diffusion equations—it significantly improves both parameter estimation accuracy and predictive reliability, particularly under low-quality, sparse observational data.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Sampling/Simulation-based Search

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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.

Problem

• Deep Learning
• Partial Differential Equations
• Parameter Estimation

Innovation

• Deep Operator Network
• Physics-Informed Neural Networks
• Bayesian Training