$PINN -- a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks

2025-04-26

AI Summary

To address the low efficiency and poor robustness in joint quantification of epistemic and aleatoric uncertainties for large-scale, multiscale partial differential equations (PDEs), this work proposes a novel integration of Bayesian physics-informed neural networks (B-PINNs) with distributed domain decomposition. For the first time, Bayesian uncertainty modeling is embedded within the conservative PINN (cPINN) framework, enforcing global solution consistency via flux continuity constraints at subdomain interfaces and enabling parallelized uncertainty inference across subdomains. On 1D and 2D multiscale PDE benchmarks, the method accurately recovers global uncertainty distributions; it remains stable under 15% observational noise and varying subdomain partitions, achieving significantly higher computational efficiency than monolithic B-PINNs. The core contribution is the development of the first distributed Bayesian uncertainty quantification (UQ) framework that simultaneously ensures physical conservation, statistical rigor, and scalability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Quantify epistemic and aleatoric uncertainties in multi-scale PDEs
• Combine local BPINNs with domain decomposition for global uncertainty
• Ensure solution continuity via flux continuity across subdomains

Innovation

• Bayesian framework for global uncertainty quantification
• Domain decomposition with flux continuity
• Concurrent local uncertainty computation