Why neural functionals suit statistical mechanics

Abstract

We describe recent progress in the statistical mechanical description of many-body systems via machine learning combined with concepts from density functional theory and many-body simulations. We argue that the neural functional theory by Sammüller et al. [Proc. Nat. Acad. Sci. 120, e2312484120 (2023)] gives a functional representation of direct correlations and of thermodynamics that allows for thorough quality control and consistency checking of the involved methods of artificial intelligence. Addressing a prototypical system we here present a pedagogical application to hard core particle in one spatial dimension, where Percus' exact solution for the free energy functional provides an unambiguous reference. A corresponding standalone numerical tutorial that demonstrates the neural functional concepts together with the underlying fundamentals of Monte Carlo simulations, classical density functional theory, machine learning, and differential programming is available online at https://github.com/sfalmo/NeuralDFT-Tutorial.

arXiv comment25 pages, 7 figures, 170 references; for associated online tutorial, see: https://github.com/sfalmo/NeuralDFT-Tutorial

References in corpus (88)

• E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials
• Dynamic Density Functional Theory of Fluids
• Dynamical density functional theory and its application to spinodal decomposition
• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Classical dynamical density functional theory: from fundamentals to applications
• Towards self-driving laboratories: The central role of density functional theory in the AI age
• Deep-Learning Density Functional Theory Hamiltonian for Efficient ab initio Electronic-Structure Calculation
• Density functional theory for hard-sphere mixtures: the White-Bear version Mark II
• Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics
• Non-equilibrium sedimentation of colloids on the particle scale
• Machine learning and density functional theory
• Machine Learning the Physical Non-Local Exchange-Correlation Functional of Density-Functional Theory
• Power functional theory for Brownian dynamics
• Computer simulation of fluid phase transitions
• Unsupervised learning for local structure detection in colloidal systems
• Colloidal Hard Spheres: Triumphs, Challenges and Mysteries
• Generalized dynamical density functional theory for classical fluids and the significance of inertia and hydrodynamic interactions
• Coarse-Grained Modelling Out of Equilibrium
• Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws
• Power functional theory for many-body dynamics
• A unified description of hydrophilic and superhydrophobic surfaces in terms of the wetting and drying transitions of liquids
• Neural-network Kohn-Sham exchange-correlation potential and its out-of-training transferability
• Primitive Model Electrolytes in the Near and Far Field: Decay Lengths from DFT and Simulations
• Quantum many-body dynamics in a Lagrangian frame: II. Geometric formulation of time-dependent density functional theory
• Predicting dynamic heterogeneity in glass-forming liquids by physics-inspired machine learning
• Use the force! Reduced variance estimators for densities, radial distribution functions and local mobilities in molecular simulations
• Noether's Theorem in Statistical Mechanics
• Neural functional theory for inhomogeneous fluids: Fundamentals and applications
• Self diffusion of particles in complex fluids: temporary cages and permanent barriers
• Scalar fundamental measure theory for hard spheres in three dimensions. Application to hydrophobic solvation
• Molecular Density Functional Theory of Water describing Hydrophobicity at Short and Long Length Scales
• Time-dependent deformation functional theory
• Better than counting: Density profiles from force sampling
• A classical density functional from machine learning and a convolutional neural network
• What buoyancy really is. A Generalized Archimedes Principle for sedimentation and ultracentrifugation
• Modified Poisson-Boltzmann equations and macroscopic forces in inhomogeneous ionic fluids
• Nonequilibrium Ornstein-Zernike relation for Brownian many-body dynamics
• Machine-learning free-energy functionals using density profiles from simulations
• Analytical classical density functionals from an equation learning network
• Thermodynamic entropy as a Noether invariant
• Velocity gradient power functional for Brownian dynamics
• Dielectric response with short-ranged electrostatics
• Machine learning many-body potentials for colloidal systems
• Density depletion and enhanced fluctuations in water near hydrophobic solutes: identifying the underlying physics
• Why Noether's Theorem applies to Statistical Mechanics
• Machine learning the derivative discontinuity of density-functional theory
• Structural and Thermodynamic Properties of Hard-Sphere Fluids
• Brownian asymmetric simple exclusion process
• A Noether Theorem for Markov Processes
• Flow and structure in nonequilibrium Brownian many-body systems
• Perspective: How to overcome dynamical density functional theory
• Force Balance Approach for Advanced Approximations in Density Functional Theories
• Structural nonequilibrium forces in driven colloidal systems
• Solitons in overdamped Brownian dynamics
• Force density functional theory in- and out-of-equilibrium
• Machine-learning effective many-body potentials for anisotropic particles using orientation-dependent symmetry functions
• Dynamic correlations in Brownian many-body systems
• Machine learning of a density functional for anisotropic patchy particles
• Physics-constrained Bayesian inference of state functions in classical density-functional theory
• Noether-Constrained Correlations in Equilibrium Liquids
• Bulk dynamics of Brownian hard disks: Dynamical density functional theory versus experiments on two-dimensional colloidal hard spheres
• Perspective: New directions in dynamical density functional theory
• Reliable emulation of complex functionals by active learning with error control
• Gravity-induced phase phenomena in plate-rod colloidal mixtures
• Single-file transport in periodic potentials: The Brownian asymmetric exclusion process
• Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids
• Force balance in thermal quantum many-body systems from Noether's theorem
• Physics-informed Bayesian inference of external potentials in classical density-functional theory
• Comparative study of force-based classical density functional theory
• Variance of fluctuations from Noether invariance
• A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density
• Thermodynamical path integral and emergent symmetry
• Thermodynamic relations at the coupling boundary in adaptive resolution simulations for open systems
• A classical density functional theory for solvation across length scales
• Statics and dynamics of inhomogeneous liquids via the internal-energy functional
• Universality in Driven and Equilibrium Hard Sphere Liquid Dynamics
• Sedimentation stacking diagram of binary colloidal mixtures and bulk phases in the plane of chemical potentials
• Exact density functional for hard rod mixtures derived from Markov chain approach
• Noether invariance theory for the equilibrium force structure of soft matter
• Density functional for hard hyperspheres from a tensorial-diagrammatic series
• Inhomogeneous steady shear dynamics of a three-body colloidal gel former
• Reduced-variance orientational distribution functions from torque sampling
• Functional-renormalization-group approach to classical liquids with short-range repulsion: a scheme without repulsive reference system
• Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids
• Nonequilibrium transport and phase transitions in driven diffusion of interacting particles
• Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory
• Classical Density Functional Theory: The Local Density Approximation
• Note on general functional flows in equilibrium systems

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• 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
• Why gauge invariance applies to statistical mechanics
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• Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures
• Metadensity functional learning for classical fluids: Regularizing with pair correlations
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• Confusion-driven machine learning of structural phases of a flexible, magnetic Stockmayer polymer
• Routes to the density profile and structural inconsistency