AI 中文总结
研究高维随机偏微分方程参数识别问题，核心方法是用增广拉格朗日乘子域分解方法优化局部神经网络模型，结合分布式分散式集合卡尔曼滤波器，主要贡献是降低计算成本，在参数估计上接近EnKF和MCMC且捕获后验参数分布。
AI 中文摘要
集合卡尔曼滤波器（EnKF）为从空间分布测量中识别基于物理定律的参数提供了一个有效框架。但其预测模型需要大量样本以准确表示不确定性，导致计算成本高。引入基于神经网络的代理模型来取代基于样本的预测模型。这种映射对于高维空间域需要大量参数。为克服此限制，开发了增广拉格朗日乘子域分解方法（DDM），在全局通信和耦合之前独立优化局部神经网络模型。这减少了神经网络参数数量，同时提高了局部逼近精度。此外，研究了基于DDM - NN代理模型的分布式和分散式集合卡尔曼滤波器方法，将参数识别问题分解为局部子问题。每个局部估计器使用本地可用信息更新材料参数，相邻子域之间的通信可重建一致的全局估计以降低计算成本。该方法在三维材料参数识别问题上进行了评估，并与EnKF和MCMC参考解决方案进行了比较。结果表明，所提出的基于DDM - NN的KF捕获了后验参数分布，接近EnKF和MCMC获得的解决方案。虽然MCMC提供了后验分布的最准确表示，但所提出的方法在降低预测模型计算要求的情况下实现了可比的参数估计。
英文摘要
Ensemble Kalman filters (EnKF) provide an efficient framework for parameter identification of physics based laws from spatially distributed measurements. Their forecast models require a large number of samples to accurately represent uncertainties, leading to high computational costs. A NN-based surrogate model is introduced to replace the sample-based forecast model. The proposed NN surrogate maps spatial coordinates and physics-based parameters to the forecasted observation. Such maps require a large number of parameters for high-dimensional spatial domains. To overcome this limitation, a augmented Lagrange multiplier domain decomposition method (DDM) is developed, where local NN models are optimized independently before global communication and coupling. This reduces the number of NN parameters while improving local approximation accuracy. Furthermore, a distributed and decentralized ensemble Kalman filter approach based on DDM-NN surrogate model is investigated, where the parameter identification problem is decomposed into local subproblems. Each local estimator updates the material parameters using locally available information, while communication between neighboring subdomains enables the reconstruction of a consistent global estimate to reduce the computational cost. The proposed method is evaluated on a three-dimensional material parameter identification problem and compared with an EnKF and a MCMC reference solution. The results show that the proposed DDM NN-based KF captures the posterior parameter distribution and approaches the solutions obtained with both EnKF and MCMC. While MCMC provides the most accurate representation of the posterior distribution, the proposed approach achieves comparable parameter estimates with reduced computational requirements for the forecast model.
