通过域分解提升物理信息神经网络的全局不确定性计算效率
$PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks
- 将局部贝叶斯PINN与域分解结合，跨子域连续性由通量守恒保证
- 在1D/2D问题中实现并行计算，全局不确定性恢复更高效
- 对噪声达15%的数据仍具鲁棒性，适合多尺度复杂PDE求解
物理信息神经网络（PINNs）是一种求解具有噪声和稀疏初始边界数据的偏微分方程（PDEs）的新方法。然而，在大规模多尺度问题中，有效量化认知不确定性和随机不确定性仍具挑战。本文提出$PINN，一种基于贝叶斯框架的域分解方法，通过组合局部贝叶斯物理信息神经网络（BPINN）来计算PDEs的全局不确定性。子域间的解连续性通过相邻子域界面的通量连续性约束实现。为验证$PINN的有效性，我们在一维和二维空间域的PDE上进行了系列数值实验。尽管采用保守型PINN（cPINNs），该方法可无缝扩展至其他域分解技术。结果表明，该方法通过精确计算各子域局部不确定性，实现了更高效的全局不确定性恢复；且在训练数据加入高达15%的不相关随机噪声、不同域尺寸条件下均表现出良好鲁棒性。
原文摘要 · Abstract (English)
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.
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