Gaussian Process Regression for Inverse Problems in Linear PDEs

2025-02-06

AI Summary

This work addresses the robust identification of physical parameters—such as wave speed—from noisy observational data in inverse problems governed by linear partial differential equations (PDEs). We propose a novel methodology that integrates algebraic analysis with Gaussian process regression: leveraging tools from commutative algebra and algebraic analysis, we construct physically consistent, structured priors; this constitutes the first algorithmic framework for generating such priors and embedding them within a Gaussian process model. The approach eliminates the subjectivity and limitations inherent in hand-crafted priors, achieving a unified balance between mathematical rigor and computational tractability. Evaluated on wave-speed inversion for the wave equation, our method achieves显著 improvements in both accuracy and computational efficiency, while demonstrating strong robustness to observational noise.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency.

Problem

• Efficient inverse problem solving
• Linear PDEs with Gaussian processes
• Wave speed identification from noise

Innovation

• Gaussian Process Regression
• Linear PDEs solution
• Macaulay2 software integration