Scalable High-Dimensional Bayesian Field Reconstruction with Finite Elements: Application to 3D Porous Media Flow

2026-05-23

AI Summary

This work addresses the challenge of high-dimensional Bayesian spatial field reconstruction—such as inverting three-dimensional permeability fields in porous media—by proposing a finite element–native variational inference framework. The approach integrates stochastic partial differential equation (SPDE) priors with a Cholesky parameterization of sparse precision matrices, enabling, for the first time, full-covariance variational reconstruction at scales exceeding 400,000 dimensions with only O(n) memory overhead. By implicitly representing the dense posterior covariance, employing pathwise derivative-based ELBO gradient estimation, adopting an automatic coarse-to-fine continuation strategy within a VB-EM scheme, and leveraging natural gradient optimization, the method substantially enhances both reconstruction accuracy and computational efficiency. Experimental results demonstrate its clear superiority over state-of-the-art methods in recovering critical spatial structures.

Abstract

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, with a probabilistic treatment of all prior and likelihood hyperparameters, on a three-dimensional curved finite-element discretization at a stochastic field dimension exceeding 400000. 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 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 O(n) memory cost and enabling direct evidence-lower-bound (ELBO) gradients via the path-derivative (sticking-the-landing) estimator. Natural gradient strategies stabilize convergence, while a variational Bayes expectation-maximization (VB-EM) loop marginalizes all hyperparameters analytically and induces an automatic coarse-to-fine continuation. The framework is demonstrated on Bayesian permeability field reconstruction for a porous-media flow problem, recovering all major spatial features with high fidelity. Algorithmic ablation and comparison with alternative inference methods quantify the improvements over state-of-the-art baselines.

Problem

• Bayesian inference
• high-dimensional reconstruction
• finite element method
• porous media flow
• spatial field

Innovation

• variational inference
• finite element method
• high-dimensional Bayesian reconstruction
• sparse precision parameterization
• stochastic PDE prior