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.

翻译：我们提出了一种统一的、以有限元为基础的变分推断框架，用于解决由偏微分方程（PDE）控制的基于物理问题中的极高维贝叶斯空间场重建，其中控制方程在推断参数上呈非线性。该框架在随机场维度超过400000的三维曲面有限元离散化上，提供了全协方差高斯变分后验，并对所有先验和似然超参数进行了概率化处理。据我们所知，这是该尺度下首次实现全协方差变分重建，补充了主导极大规模贝叶斯反演的低秩Hessian-Laplace方法。空间先验源自随机偏微分方程（SPDE）联系，并以有限元（FE）算子形式直接进行数学表述。稀疏高斯变分分布通过其精度Cholesky因子进行参数化，稀疏模式继承自域上的拉普拉斯算子。与仅编码短程相关性的基于协方差的稀疏参数化不同，稀疏精度矩阵通过其稀疏逆隐式地表示稠密的后验协方差，从而以O(n)内存成本生成光滑且物理上合理的样本，并通过路径导数（ sticking-the-landing）估计器获得直接的对数证据下界（ELBO）梯度。自然梯度策略稳定了收敛过程，而变分贝叶斯期望最大化（VB-EM）循环则通过解析方式边缘化所有超参数，并自动诱导出从粗到细的连续多尺度方法。该框架在多孔介质流动问题的贝叶斯渗透率场重建中得到了验证，高保真地恢复了所有主要空间特征。通过算法消融实验和与其他推断方法的比较，量化了相对于最先进基准方法的改进。