Scalable high-dimensional Bayesian field reconstruction with finite elements: Application to 3D porous media flow

We present a unified, finite-element-native variational inference framework for very high-dimensional Bayesian spatial field reconstruction in physics-based problems governed by partial differential equations (PDEs) that are nonlinear in the inferred parameters. The framework delivers a full-covariance Gaussian variational posterior (a dense covariance represented implicitly through a sparse precision Cholesky factor), with a probabilistic treatment of the prior and likelihood variance/precision hyperparameters, on a three-dimensional curved finite-element discretization at a stochastic field dimension exceeding 4 ⋅ 10^5. To our knowledge, this is the first full-covariance variational reconstruction at this scale, complementing the low-rank Hessian–Laplace approaches that dominate extreme-scale Bayesian inversion. The following contributions enable this: The spatial prior is derived from the stochastic PDE (SPDE) connection and formulated natively in terms of finite-element (FE) operators. The sparse Gaussian variational distribution is parameterized via its precision Cholesky factor, with the sparsity pattern inherited from the domain’s Laplacian. Unlike covariance-based sparse parameterizations, which encode only short-range correlations, the sparse precision implicitly represents dense posterior covariances through its sparse inverse, yielding smooth, physically plausible samples at memory cost and enabling direct evidence-lower-bound (ELBO) gradients via the path-derivative (sticking-the-landing) estimator, without reparameterization Jacobians. Prior, variational distribution,hyperparameter updates, and adjoint-based log-posterior gradients are all expressed in terms of the same FE operators assembled from a single discretization shared with the forward problem. Natural gradient strategies account for the ELBO curvature and stabilize convergence. All hyperparameters are marginalized analytically within a variational Bayes expectation–maximization (VB-EM) loop, which additionally induces an automatic coarse-to-fine continuation that accelerates convergence. The framework is demonstrated on Bayesian permeability field reconstruction for a porous-media flow problem on this domain, recovering all major spatial features with high fidelity. A detailed quantitative study, comprising an algorithmic ablation of every component and a comparison with alternative inference methods, provides numerical evidence of the improvements achieved by our developments relative to state-of-the-art baselines. The code for the forward problem and the inference algorithm is openly available.     «

We present a unified, finite-element-native variational inference framework for very high-dimensional Bayesian spatial field reconstruction in physics-based problems governed by partial differential equations (PDEs) that are nonlinear in the inferred parameters. The framework delivers a full-covariance Gaussian variational posterior (a dense covariance represented implicitly through a sparse precision Cholesky factor), with a probabilistic treatment of the prior and likelihood variance/precisio...     »