Physics-informed Bayesian inference of external potentials in classical density-functional theory

Abstract

The swift progression of machine learning (ML) has not gone unnoticed in the realm of statistical mechanics. ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to compute the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption and capillarity.

References in corpus (12)

• On representing chemical environments
• Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning
• Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles
• How crystals form: A theory of nucleation pathways
• Machine learning and density functional theory
• A classical density functional from machine learning and a convolutional neural network
• Geometry-induced phase transition in fluids: capillary prewetting
• Machine-learning free-energy functionals using density profiles from simulations
• Analytical classical density functionals from an equation learning network
• Perspective: How to overcome dynamical density functional theory
• Physics-constrained Bayesian inference of state functions in classical density-functional theory
• Controlling the composition of a confined fluid by an electric field

Cited by in corpus (14)

• Learning classical density functionals for ionic fluids
• Why neural functionals suit statistical mechanics
• Hyper-density functional theory of soft matter
• Neural density functionals: Local learning and pair-correlation matching
• Neural density functional theory of liquid-gas phase coexistence
• Neural force functional for non-equilibrium many-body colloidal systems
• Why hyperdensity functionals describe any equilibrium observable
• Metadensity functional theory for classical fluids: Extracting the pair potential
• Learning the bulk and interfacial physics of liquid-liquid phase separation with neural density functionals
• Chemical Physics of Controlled Wettability and Super Surfaces
• Determining the chemical potential via universal density functional learning
• Metadensity functional learning for classical fluids: Regularizing with pair correlations
• Forecasting with an N-dimensional Langevin Equation and a Neural-Ordinary Differential Equation
• Learning general pair interactions between self-propelled particles