Bayesian Variational System Identification with Weak-Form Residual Likelihoods

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.