PAC-Bayes Certificates for Bayesian Inverse Problems: A Case Study on the Heat Equation

In scientific and engineering contexts requiring inference of material properties from sparse and noisy sensor data, Bayesian inverse problems governed by partial differential equations play a central role. Although classical Bayesian methods produce credible intervals and posterior distributions, they lack finite-sample guarantees regarding forecast accuracy for new data. In safety-critical domains such as thermal engineering, materials testing, and structural health monitoring, too much faith on uncertainty estimates can result in inaccurate predictions and potentially terrible outcomes. This study addresses the identified gap by introducing the first Probably Approximately Correct-Bayesian (PAC-Bayes) generalization certificates for Bayesian inverse partial differential equation (PDE) problems, using thermal conductivity inference in the one-dimensional heat equation as a case study. The suggested methodology offers finite-sample, distribution-free upper limits on prediction error by utilizing the connection between Gibbs distributions and tempered Bayesian posteriors.Theoretical validity is maintained through the use of a sigmoid-bounded squared loss function, which preserves sensitivity to prediction quality. To ensure certificate validity across varying mesh resolutions and to achieve monotonic improvement with refinement, the approach incorporates a mesh-robust decomposition that separates statistical generalization error from numerical discretization bias. Extensive experiments involving 1,728 parameter combinations systematically vary mesh resolutions, posterior temperatures, noise levels, and sensor counts. Results demonstrate that PAC-Bayes certificates effectively identify overfitting in cases where conventional credible intervals are misleadingly narrow, particularly under low sensor density or high noise conditions, where reliability is essential. Certificate gaps, typically between 7-9%, provide conservative and practical bounds that are independent of mesh artifacts. The discretization penalty decreases with secondorder convergence, supporting the robustness of the statistical guarantee. Ongoing certificate extensions enable applicability in streaming and iterative computational contexts. The proposed architecture functions as a modular post-inference layer that integrates into any Bayesian inverse partial differential equation pipeline without modification of priors, likelihoods, or solvers. By offering auditable and conservative generalization guarantees that complement parameter-space credible intervals, this approach enhances the reliability and credibility of uncertainty quantification in scientific machine learning and data-limited engineering contexts, particularly where decision-making carries significant consequences. The complete methodology, including code and reproducibility artifacts, is publicly available.

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