Bayesian Inference for Initial Heat States with Gaussian Series Priors

2025-06-17

AI Summary

This paper addresses the Bayesian inverse problem of recovering the initial temperature field for the heat equation in heterogeneous media, using noisy interior observations at a fixed time. We propose a Gaussian series prior expansion in the eigenfunctions of the Dirichlet–Laplacian operator—a novel systematic incorporation of this prior into heat equation initial-value inverse problems—yielding a conjugate posterior with closed-form analytic expression. The method ensures theoretical interpretability and asymptotic optimality: the posterior mean achieves the minimax optimal rate in nonparametric statistics; simultaneously, it delivers both point estimation and rigorous uncertainty quantification. Numerical experiments demonstrate that the proposed approach significantly outperforms existing methods in robustness to observational noise and calibration of posterior uncertainty.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Probabilistic InferenceSearch and Optimization: Heuristic Search

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Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider the statistical linear inverse problem of recovering the unknown initial heat state from noisy interior measurements over an inhomogeneous domain of the solution to the heat equation at a fixed time instant. We employ nonparametric Bayesian procedures with Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis, yielding convenient conjugate posterior distributions with explicit expressions for posterior inference. We review recent theoretical results that provide asymptotic performance guarantees (in the large sample size limit) for the resulting posterior-based point estimation and uncertainty quantification. We further provide an implementation of the approach, and illustrate it via a numerical simulation study.

Problem

• Recover initial heat state from noisy measurements
• Use Bayesian inference with Gaussian series priors
• Provide asymptotic guarantees for estimation accuracy

Innovation

• Bayesian inference with Gaussian series priors
• Conjugate posterior distributions for heat state
• Asymptotic guarantees for posterior estimation