We consider system identification for discovering parameterized operators in governing partial differential equations (PDEs) from noisy spatiotemporal data. Building on variational system identification (VSI), which identifies PDEs through Galerkin weak-form residuals, we develop a Bayesian VSI (B-VSI) framework for operator selection, parameter estimation, and uncertainty quantification. The central idea is to define the likelihood directly in weak-form residual space by propagating observation uncertainty through the weak-form residual map. The resulting likelihood captures heteroscedastic and correlated residual errors while avoiding repeated forward PDE solves during inference. For efficient computation, we use lagged-covariance updates that yield generalized least-squares estimates and conjugate posterior approximations when applicable, together with gradient-based and particle-based methods for more general priors and posterior structures. Model-form uncertainty is handled through sequential operator elimination guided by a residual-space Bayesian information criterion. We demonstrate the framework on state-linear and nonlinear PDEs, including the Fokker--Planck equation and a two-field Cahn--Hilliard equation. The results show that B-VSI accurately recovers active operators and coefficients from noisy data, improves robustness relative to classical VSI, and provides posterior uncertainty estimates for coefficients and derived physical quantities.

翻译：摘要：本文针对含噪时空数据中控制偏微分方程的参数化算子识别问题，提出了一种贝叶斯变分系统辨识框架。该框架在经典变分系统辨识（通过伽辽金弱形式残差识别偏微分方程）的基础上，实现了算子选择、参数估计与不确定性量化的统一。核心思想是：通过将观测不确定性经弱形式残差映射传播，直接在弱形式残差空间定义似然函数。所得似然函数可捕捉异方差相关残差误差，同时避免推理过程中的正向偏微分方程求解。在计算效率方面，我们采用滞后协方差更新技术，在适用条件下生成广义最小二乘估计与共轭后验近似，并结合基于梯度与粒子的方法处理更通用的先验及后验结构。通过基于残差空间贝叶斯信息准则的序贯算子消除策略处理模型形式不确定性。我们在状态线性/非线性偏微分方程（包括福克-普朗克方程与双场卡恩-希利亚德方程）上验证了该框架。结果表明：贝叶斯变分系统辨识能从含噪数据中准确恢复活跃算子及其系数，相较于经典变分系统辨识具有更强的鲁棒性，并能给出系数及衍生物理量的后验不确定性估计。