用深度算子网络融合物理约束，实现偏微分方程参数的鲁棒贝叶斯估计。
Deep Operator Networks for Bayesian Parameter Estimation in PDEs
- 结合DeepONet与物理信息神经网络，融合数据与物理规律建模。
- 在噪声数据和缺失方程下仍可准确估计参数并量化不确定性。
- 适用于正反问题求解，特别适合稀疏观测场景。
我们提出一种新框架，将深度算子网络（DeepONets）与物理信息神经网络（PINNs）结合，用于求解偏微分方程（PDEs）并估计其未知参数。通过融合数据驱动学习与物理约束，该方法在多种场景下均能实现鲁棒且精确的求解。采用变分推断实现贝叶斯训练，全面量化了认知不确定性与偶然不确定性。即使在噪声环境或部分物理方程缺失时，也能保证可靠预测与参数估计。该框架在求解一维非稳态热传导方程、二维反应-扩散方程以及稀疏噪声观测下的回归任务中表现出色。本方法为偏微分方程代理模型中的不确定性量化提供了一种计算高效且通用的解决方案。
原文摘要 · Abstract (English)
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.
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