Uncertainty Quantification in PDEs with Physics-Informed Neural Networks via Bayesian Domain Decomposition

In a study published on April 26, 2025, titled PINN — a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks, scientists present an advanced technique combining Bayesian methods with domain decomposition to solve partial differential equations more efficiently by addressing both local and global uncertainties.

The study introduces a novel method combining Bayesian Physics-Informed Neural Networks (BPINN) with domain decomposition to quantify global uncertainty in solving partial differential equations (PDEs). By enforcing flux continuity across subdomains, the approach ensures solution continuity while efficiently recovering global uncertainty through concurrent local uncertainty quantification. Tested on 1D and 2D PDEs, the method demonstrates robustness under up to 15% uncorrelated random noise in training data, validating its effectiveness for multi-scale problems with sparse or noisy initial/boundary conditions.

Partial differential equations (PDEs) are foundational to modeling natural phenomena across scientific disciplines, from fluid dynamics to climate science. Traditional methods for solving PDEs often encounter challenges with accuracy and scalability, particularly when dealing with complex domains or large-scale problems. Enter physics-informed neural networks (PINNs), a modern approach that leverages machine learning to solve these equations more effectively. As research advances, new innovations are emerging to address the limitations of traditional PINNs, paving the way for enhanced computational tools.

One notable innovation is the development of Finite Basis Physics-Informed Neural Networks (FBPINNs). These networks employ domain decomposition, breaking down complex problems into smaller, more manageable parts. By integrating finite basis functions, FBPINNs improve scalability and efficiency, making them particularly effective for large-scale domains. This method enhances precision in handling intricate geometries and boundary conditions, significantly improving the accuracy of solutions.

Building on this foundation, Augmented Physics-Informed Neural Networks (APINNs) introduce a novel approach through gating networks. These networks enable soft domain decomposition, allowing the model to adapt dynamically to problem complexity without being constrained by fixed boundaries. This flexibility is especially beneficial in scenarios involving irregular or evolving geometries, offering enhanced precision and robustness compared to traditional methods.

Another advancement comes in the form of Extended Physics-Informed Neural Networks (XPINNs). These networks focus on improving generalization by extending the loss function, ensuring that solutions remain consistent across various scales and domains. This capability is crucial for tackling inverse problems, where the goal is to determine unknown parameters from observed data. XPINNs are particularly valuable in fields like engineering and environmental modeling, where such problems are common.

The evolution of PINNs into FBPINNs, APINNs, and XPINNs represents a significant advancement in solving PDEs. These innovations enhance computational efficiency and accuracy while opening new possibilities for addressing complex real-world challenges. From optimizing industrial designs to improving climate models, the impact of these advanced neural networks is vast and transformative. As research continues, further refinements are expected, pushing the boundaries of what is achievable in computational science.