$PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks (2504.19013v3)

Abstract: Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric uncertainties in big multi-scale problems remains challenging. We propose \$PINN a novel method of computing global uncertainty in PDEs using a Bayesian framework, by combining local Bayesian Physics-Informed Neural Networks (BPINN) with domain decomposition. The solution continuity across subdomains is obtained by imposing the flux continuity across the interface of neighboring subdomains. To demonstrate the effectiveness of \$PINN, we conduct a series of computational experiments on PDEs in 1D and 2D spatial domains. Although we have adopted conservative PINNs (cPINNs), the method can be seamlessly extended to other domain decomposition techniques. The results infer that the proposed method recovers the global uncertainty by computing the local uncertainty exactly more efficiently as the uncertainty in each subdomain can be computed concurrently. The robustness of \$PINN is verified by adding uncorrelated random noise to the training data up to 15% and testing for different domain sizes.

• The paper introduces the $PINN framework, integrating Bayesian methods for uncertainty quantification and domain decomposition for scalability into Physics-Informed Neural Networks to solve partial differential equations.
• $PINN demonstrates superior performance over traditional PINNs, effectively quantifying uncertainty, handling noisy data (up to 15% noise), and improving computational efficiency through decomposition.
• This framework offers a robust approach for scientific machine learning, enabling more reliable solutions for complex, multi-scale problems with limited or uncertain data by combining uncertainty handling and scalability.

Overview of the \$PINN Framework for Bayesian Physics-Informed Neural Networks</h2> <p>The paper introduces a novel computational framework called \$PINN, designed to enhance the performance and scalability of Physics-Informed Neural Networks (PINNs) in solving partial differential equations (PDEs), particularly under conditions of noisy and sparse data. The primary advancement presented in this framework is the integration of a Bayesian approach with domain decomposition, aiming to address uncertainty quantification and scalability issues that traditional PINNs face.

Physics-Informed Neural Networks (PINNs)

PINNs are known for their ability to incorporate physical laws within neural network architectures, enabling them to approximate solutions to PDEs by combining data-driven learning with fundamental physical principles. These networks have shown promise in addressing both forward and inverse problems by embedding the constraints dictated by differential equations directly into their loss functions. Nonetheless, their performance can degrade when confronted with noisy data or when tasked with solving complex, multi-scale problems.

Bayesian Physics-Informed Neural Networks (BPINNs)

The addition of a Bayesian framework to PINNs provides a mechanism for uncertainty quantification (UQ) that PINNs lack. BPINNs use Bayesian inference to handle uncertainties, distinguishing between epistemic uncertainties (arising from model limitations or parameter variability) and aleatoric uncertainties (related to inherent data noise). The Bayesian component involves modeling uncertainties using probability distributions over model parameters and iteratively updating these distributions through observed data, typically utilizing methods such as Hamiltonian Monte Carlo (HMC) or variational inference.

Domain Decomposition with PINNs

Domain decomposition is a critical technique when solving large-scale PDE problems, as it divides the computational domain into smaller, manageable subdomains. This technique not only facilitates parallel computation but also enhances the flexibility of PINNs to tackle problems involving complex geometries or coupling multiple physical phenomena. Approaches like conservative PINNs (cPINNs) illustrate how boundary and interface conditions can be imposed between subdomains to maintain solution continuity.

The \$PINN Framework</h3> <p>The \$PINN framework synthesizes the Bayesian approach with domain decomposition, offering a comprehensive solution that leverages the strengths of BPINNs and cPINNs. It enforces solution continuity across subdomains and efficiently calculates global uncertainties by estimating local uncertainties within subdomains concurrently. This is achieved by computing flux continuity across interfaces, ensuring accuracy and robustness even under conditions of significant noise (up to 15% in the experiments). Moreover, \$PINN supports both forward problems, where the solution is sought for known parameters, and inverse problems, where unknown parameters are inferred from data.</p> <h3 class='paper-heading'>Key Results and Applications</h3> <p>The paper illustrates the efficacy of \$PINN through various computational experiments involving PDEs in one-dimensional and two-dimensional spatial domains. \$PINN demonstrates superior performance in managing uncertainty and optimizing computational efficiency relative to traditional PINNs. The inclusion of a Bayesian framework not only aids in uncertainty quantification but also helps avoid overfitting—a common challenge when dealing with noisy data.</p> <h3 class='paper-heading'>Implications and Future Directions</h3> <p>\$PINN represents a significant advancement in the application of neural networks to scientific computing, particularly in scenarios where data is limited or characterized by high levels of uncertainty. By accommodating both domain decomposition and a Bayesian approach, \$PINN offers a robust framework capable of handling complex, multi-scale problems, potentially paving the way for more widespread adoption of scientific machine learning techniques in engineering and physics. Future research may focus on further optimization of the algorithm for parallel computing environments and extending its application to a broader range of PDEs and problem domains, including higher dimensions and real-world scenarios.</p> <p>In summary, the \$PINN framework presents a promising paradigm for enhancing the capability of neural networks in addressing scientific challenges by integrating uncertainty quantification and computational scalability, ensuring reliable and efficient problem-solving in complex systems.