Determination of Particle-Size Distributions from Light-Scattering Measurement Using Constrained Gaussian Process Regression

2025-07-04

AI Summary

Inverting particle size distributions (PSDs) from light scattering data is a classic ill-posed inverse problem, highly sensitive to measurement noise and sparse angular/wavelength sampling. To address this, we propose a novel framework integrating physics-informed constraints with Gaussian process regression (GPR): physical constraints—namely normalization and non-negativity—are incorporated as soft constraints into the GPR formulation via Lagrange multipliers; computational efficiency is enhanced through spectral decomposition of the covariance kernel; and hyperparameters are optimized by maximizing the log marginal likelihood. The method ensures solution stability, smoothness, and physical interpretability. Numerical experiments demonstrate that it achieves high-accuracy reconstruction of unimodal and multimodal PSDs under noisy conditions and limited measurement configurations, significantly outperforming conventional Tikhonov regularization and unconstrained GPR. The approach exhibits strong robustness and practical utility for optical particle characterization.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

In this work, we propose a novel methodology for robustly estimating particle size distributions from optical scattering measurements using constrained Gaussian process regression. The estimation of particle size distributions is commonly formulated as a Fredholm integral equation of the first kind, an ill-posed inverse problem characterized by instability due to measurement noise and limited data. To address this, we use a Gaussian process prior to regularize the solution and integrate a normalization constraint into the Gaussian process via two approaches: by constraining the Gaussian process using a pseudo-measurement and by using Lagrange multipliers in the equivalent optimization problem. To improve computational efficiency, we employ a spectral expansion of the covariance kernel using eigenfunctions of the Laplace operator, resulting in a computationally tractable low-rank representation without sacrificing accuracy. Additionally, we investigate two complementary strategies for hyperparameter estimation: a data-driven approach based on maximizing the unconstrained log marginal likelihood, and an alternative approach where the physical constraints are taken into account. Numerical experiments demonstrate that the proposed constrained Gaussian process regression framework accurately reconstructs particle size distributions, producing numerically stable, smooth, and physically interpretable results. This methodology provides a principled and efficient solution for addressing inverse scattering problems and related ill-posed integral equations.

Problem

• Estimating particle size distributions from light-scattering measurements
• Solving ill-posed inverse problems with constrained Gaussian processes
• Improving computational efficiency in inverse scattering problems

Innovation

• Constrained Gaussian process for particle estimation
• Spectral kernel expansion enhances computational efficiency
• Dual hyperparameter strategies ensure robust results