Bayesian Variational System Identification with Weak-Form Residual Likelihoods

2026-06-12

AI Summary

This work proposes a Bayesian framework based on variational system identification (VSI) to address the identification of partial differential equation (PDE) governing operators and quantification of their uncertainties in noisy spatiotemporal data. The method constructs the likelihood function in the weak-form residual space for the first time, effectively accommodating heteroscedastic and correlated errors, and integrates a sequential operator elimination strategy to handle model-form uncertainty. By leveraging weak-form residual mapping, lagged covariance updates, generalized least squares estimation, and conjugate posterior approximations, it synergistically combines gradient-based and particle-based Bayesian inference while avoiding repeated forward PDE solves. Numerical experiments on the Fokker–Planck and two-field Cahn–Hilliard equations demonstrate that the approach accurately recovers active operators and parameters, exhibits superior robustness over classical VSI, and yields reliable posterior uncertainty estimates for both parameters and derived physical quantities.

Abstract

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.

Problem

• system identification
• partial differential equations
• operator discovery
• noisy spatiotemporal data
• uncertainty quantification

Innovation

• Bayesian system identification
• weak-form residuals
• uncertainty quantification
• variational inference
• PDE discovery