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Physics-Informed Gaussian Process Inference of Liquid Structure from Scattering Data

Harry W. Sullivan, Brennon L. Shanks, Matej Cervenka, Michael P. Hoepfner

TL;DR

This work tackles the challenge of extracting physically meaningful RDFs from scattering data with quantified uncertainty. The authors develop a physics-informed, non-stationary Gaussian process framework that places a GP prior over the structure factor $S(q)$ and propagates uncertainty through the linear radial Fourier transform to obtain a probabilistic RDF $g(r)$, while enforcing key boundary behaviors via a Gibbs kernel and a bonded/non-bonded mean. Hyperparameters are learned by maximizing the log marginal likelihood with automatic differentiation, and coordination numbers are derived with uncertainty via Monte Carlo sampling of the RDF posterior. The method is validated on liquid argon and water, including simulated and experimental data, yielding credible RDFs, peak statistics, and coordination numbers that align with known benchmarks and offer principled uncertainty quantification. This approach provides a transparent, model-based uncertainty framework that can benchmark molecular simulations and connect scattering data to thermodynamic properties and interatomic potentials.

Abstract

We present a nonparametric Bayesian framework to infer radial distribution functions from experimental scattering measurements with uncertainty quantification using non-stationary Gaussian processes. The Gaussian process prior mean and kernel functions are designed to mitigate well-known numerical challenges with the Fourier transform, including discrete measurement binning and detector windowing, while encoding fundamental yet minimal physical knowledge of liquid structure. We demonstrate uncertainty propagation of the Gaussian process posterior to unmeasured quantities of interest. The methodology is applied to liquid argon and water as a proof of principle. The full implementation is available on GitHub at https://github.com/hoepfnergroup/LiquidStructureGP-Sullivan.

Physics-Informed Gaussian Process Inference of Liquid Structure from Scattering Data

TL;DR

This work tackles the challenge of extracting physically meaningful RDFs from scattering data with quantified uncertainty. The authors develop a physics-informed, non-stationary Gaussian process framework that places a GP prior over the structure factor and propagates uncertainty through the linear radial Fourier transform to obtain a probabilistic RDF , while enforcing key boundary behaviors via a Gibbs kernel and a bonded/non-bonded mean. Hyperparameters are learned by maximizing the log marginal likelihood with automatic differentiation, and coordination numbers are derived with uncertainty via Monte Carlo sampling of the RDF posterior. The method is validated on liquid argon and water, including simulated and experimental data, yielding credible RDFs, peak statistics, and coordination numbers that align with known benchmarks and offer principled uncertainty quantification. This approach provides a transparent, model-based uncertainty framework that can benchmark molecular simulations and connect scattering data to thermodynamic properties and interatomic potentials.

Abstract

We present a nonparametric Bayesian framework to infer radial distribution functions from experimental scattering measurements with uncertainty quantification using non-stationary Gaussian processes. The Gaussian process prior mean and kernel functions are designed to mitigate well-known numerical challenges with the Fourier transform, including discrete measurement binning and detector windowing, while encoding fundamental yet minimal physical knowledge of liquid structure. We demonstrate uncertainty propagation of the Gaussian process posterior to unmeasured quantities of interest. The methodology is applied to liquid argon and water as a proof of principle. The full implementation is available on GitHub at https://github.com/hoepfnergroup/LiquidStructureGP-Sullivan.

Paper Structure

This paper contains 27 sections, 52 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theory and Methods
  3. The Gaussian Process Model
  4. Posterior Estimation of the Radial Distribution Function
  5. Designing a Gaussian Process for Liquid Structure Factors
  6. Non-Stationary Kernel Selection
  7. Mean Selection
  8. Hyperparameter Optimization
  9. Coordination Number Analysis
  10. Results and Discussion
  11. Liquid Argon
  12. Water
  13. Simulated Liquid Water
  14. Experimental X-ray Scattering of Liquid Water
  15. Discussion
  16. ...and 12 more sections

Figures (14)

  • Figure 1: A flowchart corresponding to the GPFT algorithm applied to scattering data.
  • Figure 2: Gaussian process kernels after hyperparameter fitting of argon at temperature $T=85$[K] and density $\rho = 0.02125 \ [\text{atom}/\angstrom^3]$. Left corresponds to eq \ref{['eq:Krr']}, middle corresponds to eq \ref{['eq:Krq']}, and right corresponds to eq \ref{['eq:Kqq']}. The colorbar represents the range of values indicated by the colormap. Any values outside the specified range are clipped and displayed using the colors corresponding to the nearest boundary.
  • Figure 3: Posterior of the Gaussian process fit to argon at temperature $T=85$[K] and density $\rho = 0.02125 \ [\text{atom}/\angstrom^3]$. (a) Posterior covariance in $q$-space from eq \ref{['eq:SQ_post_sigma']}. (b) Prior and posterior distribution for the argon structure factor from eq \ref{['eq:SQ_post_mu']}. (c) Posterior covariance in $r$-space from eq \ref{['eq:RDF_post_sigma']}. (d) Prior and posterior argon RDF from eq \ref{['eq:RDF_post_mu']}. The colorbars are clipped similarly to Figure \ref{['fig:kernelviz']}.
  • Figure 4: Posterior of the Gaussian process fit to structure factors derived from NVT simulations at a temperature of 298.15 K and density 1 g cm$^{-3}$ with flexible TIP4P/2005f water. (a) Posterior covariance in $q$-space from eq \ref{['eq:SQ_post_sigma']}. (b) Prior and posterior distribution for the hydrogen-hydrogen structure factor from eq \ref{['eq:SQ_post_mu']}. (c) Posterior covariance in $r$-space from eq \ref{['eq:RDF_post_sigma']}. (d) Prior and posterior hydrogen-hydrogen RDF from eq \ref{['eq:RDF_post_mu']}.
  • Figure 5: Posterior of the Gaussian process fit to the X-ray scattering data. (a) Posterior covariance in $q$-space from eq \ref{['eq:SQ_post_sigma']}. (b) Prior and posterior distribution for the oxygen-oxygen structure factor from eq \ref{['eq:SQ_post_mu']}. (c) Posterior covariance in $r$-space from eq \ref{['eq:RDF_post_sigma']}. (d) Prior and posterior oxygen-oxygen RDF from eq \ref{['eq:RDF_post_mu']}. (e) GP Mean subtracted comparison between the uncertainty estimates from the non-stationary GP approach and Skinner's interpretationskinner_benchmark_2013.
  • ...and 9 more figures