A structure-preserving neural density functional for the ions of a polymer electrolyte

The gist Predicting the structure and response of inhomogeneous polymer electrolytes requires a description of ion correlations that retains molecular-scale accuracy while remaining transferable across spatial scales and geometries

Theory

The paper develops a neural density functional for the ions of a coarse-grained salt-doped polymer melt that preserves spatial symmetries, thermodynamic integrability and the Noether identities, with perfect screening recovered in stable, noncritical bulk states The functional is built from an analytical Coulomb interaction and a scalar functional of the cation and anion densities alone The excess part of the free-energy functional, denoted as Fex[n], is parameterized by radial convolution kernels and a pointwise neural network evaluated on weighted densities The chemical potentials are defined as functional derivatives of the same scalar free-energy functional, ensuring that free-energy differences between density profiles do not depend on the integration path The functional is invariant under all translations, rotations, and reflections because all kernels are radial and the pointwise function acts on weighted densities Noether’s theorem requires that the total internal force and torque vanish for any admissible density profile because the functional is invariant under these transformations The functional satisfies the perfect-screening sum rules in stable, noncritical homogeneous bulk states because the explicit Coulomb term fixes small-wavenumber charge response while the short-ranged learned contribution has a finite limit as k → 0

How it works

The neural functional is trained by matching species-resolved internal force densities measured in canonical molecular dynamics under static planar external potentials The loss function minimizes the weighted squared residuals and the weighted squared Fourier amplitudes of the residuals, defined as L (theta) = sum over runs r alpha sum over bins i=1 to Nr wr alpha i Delta squared r alpha i + sum over runs r alpha wbar r alpha floor(N/2) m=1 2 Nr bDelta r alpha(km)2 (km sigma) squared This force-based supervision differs from fitting bulk pair correlations or fitting free energies obtained by thermodynamic integration The parameters refer to no grid or box size, allowing the trained functional to be evaluated in domains of other sizes and geometries

Predictions and Transferability

The functional predicts ionic structure in larger domains and in two-dimensional external fields when trained solely on planar density and internal-force profiles from molecular dynamics It describes density profiles at an untrained salt concentration and predicts bulk structure factors and the long-wavelength number response The functional is more accurate than three other neural density-functional architectures and keeps its accuracy with a quarter of the training runs, where the errors of the best alternative grow by about two thirds The spatial transferability provides a necessary foundation for connecting molecular correlations to continuum predictions at larger scales

Bulk Correlations and Number Response

In a uniform bulk state, density fluctuations are determined by the Hessian of the free energy, which is not used in training The Fourier-space excess kernel is defined as Walpha beta (k) = integral dr exp-ik·r delta squared Fex d nalpha(r) delta nbeta(0) Including the ideal-gas and Coulomb contributions, the Ornstein–Zernike relation gives S(k)−1 = Nbar−1 + 4pi lB/k squared zzT + W(k) kBT The neural functional reproduces the zero-field MD charge and number structure factors with relative root-mean-square errors of 2.7–4.1% over 0.25 ≤ k sigma ≤ 2.1 The long-wavelength number response Γ(k) is reproduced, showing enhanced number fluctuations at low concentration and suppressed fluctuations at high concentration

Comparison of Learned Architectures

The paper compares the learned functional with a one-body network that predicts excess chemical potentials directly from local density windows, a convolutional free energy that maps the whole planar profile to a free energy, and a lattice free energy built from Cartesian moments of local density environments The neural functional has the lowest held-out errors with the full training set, 1.50% at εr = 7.5 and 3.11% at εr = 2 The one-body network's errors grow by about two thirds when the training set is reduced to a quarter, whereas the neural functional's errors change little

Discussion

The constructed neural density functional combines integrability, continuous spatial symmetry, the Noether identities and an explicit treatment of long-range electrostatics with parameters that refer to no grid Trained only on planar force densities, the same functional predicts ionic structure in larger boxes and in fields that vary in two directions A natural next step is to apply the functional to transport by supplying the chemical potentials whose gradients drive ion fluxes The range of validity out of equilibrium is set by how close the unresolved degrees of freedom remain to their equilibrium distribution

Data Availability

The project repository is https://github.com/Lyuliyao/NDFT-polymer-electrolyte, which contains analysis and training code, processed molecular-dynamics profiles, model configurations, saved metrics and selected trained parameters The comparison in the main text uses these same fits The structural checks on the MD profiles show the differences of Table 1 in the adapted implementations (Table 2) The one-body network has net-force errors of up to 14%, Jacobian asymmetries of up to 55% and path-dependent free-energy differences of up to 1% The convolutional free energy cannot be evaluated because its read-out fixes the number of grid points The Lennard-Jones control at T = 1.5 gives the same ordering: our error is 0.27% with all runs and 0.

1. Improved ion correlation prediction in inhomogeneous polymer electrolytes: The neural density functional can predict ionic structure in larger domains and in two-dimensional external fields without retraining, providing a necessary foundation for connecting molecular correlations to continuum predictions at larger scales.

2. Enhanced concentration-dependent correlation capture: The functional's nonlinear density dependence captures the concentration-dependent correlations missed by a fixed quadratic pair closure, including the crucial crossover from enhanced to suppressed long-wavelength number fluctuations at strong coupling.

3. Transferability across geometries and box sizes: Because no parameter refers to a grid or a box size, the functional can be evaluated in domains of other sizes and geometries and shows high accuracy in larger boxes (e.g., four times the training length) for both planar and nonplanar potentials.

4. Accurate bulk structure factor prediction: The functional accurately predicts bulk structure factors with relative root-mean-square errors between 2.7% and 4.1% at various coupling strengths, outperforming three other neural density-functional architectures in the training data set.

5. Improved long-wavelength number response analysis: The functional reproduces the concentration dependence of enhanced number fluctuations at low concentration and suppressed fluctuations at high concentration by accurately predicting the factor Γ(k1) = 2¯n/SNN(k1) across different dielectric constants.