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

• Ion correlations complicate electrolyte prediction because salt concentration changes pairing and clustering, while correlations span both polymer-scale and shorter packing and solvation lengths.The introduction calls for continuum theory retaining molecular fidelity across concentrations and external fields.Source: §1 · p. 1
• Earlier learned correlations need not derive from a free-energy functional and may respect only training-grid translations, whereas prior invariant approaches lacked the combined electrostatics and geometry transfer sought here.The present work trains by matching internal forces in inhomogeneous environments without chemical-potential labels.Source: §1 · p. 2
• The functional combines integrability, continuous translation, rotation, and reflection symmetry, Noether identities, explicit long-range electrostatics, and grid-independent parameters.Its construction recovers perfect screening in stable, noncritical bulk states.Source: §1 · p. 2
• The functional outperforms three neural architectures on the same ion data and retains accuracy with a quarter of the runs, while the best alternative’s errors grow by about two thirds.Source: §1 · pp. 2–3

2.1 Free-energy formulation

• At fixed particle numbers, equilibrium ion profiles minimize the intrinsic free energy plus external-potential contributions; its excess term captures correlations beyond ideal-gas and Coulomb mean-field contributions.With the polymer integrated out, the excess free energy includes chain response and refers to one host state and coupling strength.Source: §2.1 · p. 3

2.2 The neural functional

• The excess functional combines density-independent radial pair kernels with a pointwise neural network acting on weighted densities, making effective correlations nonlinear and the representation grid-independent.The radial kernels probe profiles over multiple length scales.Source: §2.2 · p. 4

2.3 Properties of the neural functional

• The scalar functional guarantees integrability and symmetric Hessians; spatial invariance yields zero total internal force and torque, while explicit Coulomb interactions ensure perfect screening in stable, noncritical bulk states.Source: §2.3 · p. 4

2.4 Training by force matching

• Training minimizes weighted residuals between MD and functional internal force densities; taking chemical-potential gradients removes constant offsets, so force matching needs no chemical-potential labels.Source: §2.1 · p. 4, §2.4 · p. 5

𝑟Í𝑖𝑤𝑟𝛼𝑖(Methods).

• The simulated system contains 400 chains of 30 beads and 120–960 ion pairs, with two dielectric constants and state points equilibrated at zero pressure.Source: 𝑟Í𝑖𝑤𝑟𝛼𝑖(Methods) · p. 5

3.2 External potentials and profiles

• Four potential families span several length scales, and each run records density and internal-force profiles in 0.1σ bins after equilibration and sampling.The families include Fourier modes, periodic Gaussian wells, and long-wavelength modes.Source: §3.2 · p. 5

3.3 Neural functional and training

• Six learned radial kernels parameterize the pair interactions, while a pointwise network acts on 12 weighted densities; the full functional has 19,641 parameters.The quadratic pair closure omits the pointwise network and has 1,336 parameters; training uses AdamW with early stopping on validation residuals.Source: §3.3 · p. 5

3.4 Other learned functionals

• Three comparator architectures are adapted to the two ion species and trained with the same data, splits, and force-matching loss: one-body, lattice-free-energy, and convolutional-free-energy models.Source: §3.4 · pp. 5–6

4.1 Density profiles and concentration transfer

• At the untrained concentration c=0.05, mean relative L2 profile errors are 0.85% and 0.98% at εr=7.5 and 2, respectively.Across all test runs, errors are 1.2% and 2.0%, compared with 3.0% and 5.7% for the pair closure and 25% and 23% for Poisson–Boltzmann theory.Source: Figure 2 · p. 6, §4.1 · p. 7

4.2 Transfer across domain sizes and geometries

• In boxes twice the training length, profile errors under an aperiodic potential are 0.8%–1.6%, versus 2.4%–6.2% for the pair closure.At four times the training length, errors are 0.5%–1.3%, versus 0.8%–2.5% for the pair closure.Source: §4.2 · p. 7
• For aperiodic two-dimensional fields, density-map errors are 1.8%–3.1%, at the level of MD noise of 1.7%–3.0%, versus 3.6%–5.5% for the pair closure.For periodic potentials, errors are 0.6%–3.7%, versus 2.4%–7.6% for the pair closure.Source: §4.2 · p. 7, Figure 3 · p. 8

4.3 Bulk correlations and number response

• At εr=7.5 and 2, the functional reproduces bulk charge and number structure factors with relative root-mean-square errors of 2.7%–4.5% and 2.4%–5.3%, respectively.The reported ranges cover 0.25 ≤ kσ ≤ 2.1 and concentrations c=0.02 to 0.08.Source: §4.3 · p. 7, Figure 4 · p. 9
• At εr=2, MD Γ(k1) rises from 0.42 ± 0.07 at c=0.01 to 11.1 ± 0.8 at c=0.08, and the functional reproduces the concentration dependence missed by the pair closure.The response describes enhanced number fluctuations at low concentration and suppressed fluctuations at high concentration.Source: §4.3 · p. 7, Figure 4 · p. 9
• The number-channel kernel changes sign with concentration at εr=2, and the functional reproduces this dependence, unlike the density-independent pair closure.The passage relates a negative kernel to enhanced number fluctuations and a positive kernel to suppressed fluctuations.Source: §4.3 · p. 8, Figure 5 · p. 9

4.4 Comparison of learned architectures

• 1.50% and 3.11% are the functional’s lowest held-out errors with full training at εr=7.5 and 2, and its errors change little with one-quarter of the runs.With one-quarter of the runs, errors are 1.55% and 3.56%; the one-body network’s errors grow by about two thirds, to 3.00% and 5.55%.Source: §4.4 · pp. 8–9
• The functional has a symmetric Hessian, path-independent free-energy differences to numerical precision, and maximum net-force error of 2 × 10^-6.On resampled grids, its chemical potentials change by less than 10^-4; the adapted baselines show larger structural deviations.Source: §4.4 · p. 10

5 Discussion

• The functional predicts ionic structure in larger boxes and two-directional fields, describes an untrained salt concentration, and recovers bulk structure factors and long-wavelength number response.Its nonlinear density dependence captures concentration-dependent correlations, including a strong-coupling sign change in the number-channel kernel that a density-independent pair kernel cannot represent.Source: §5 · p. 10
• The functional combines integrability, continuous spatial symmetry, Noether identities, and explicit long-range electrostatics, with parameters independent of any grid.The discussion proposes applying its chemical potentials to transport theories and studying electrode interfaces once boundary conditions and the polymer environment are specified.Source: §5 · p. 10
• The benchmark reports profile errors and structural checks, while excluding failed solutions from means and identifying some nonconvergence and density-collapse cases.Failure counts vary with dielectric setting and training fraction.Source: Table 2 · p. 10, §5 · p. 10

Data Availability

• The project repository contains analysis and training code, processed molecular-dynamics profiles, model configurations, saved metrics, and selected trained parameters.Source: Data Availability · p. 11

S1. Properties that hold by construction

• The functional is invariant under translations, rotations, and reflections, so its forces transform covariantly without requiring an equivariant architecture.The proof derives invariance from scalar-functional construction and radial kernels.Source: S1. Properties that hold by construction · pp. 13–14
• The bulk interaction kernel is real, isotropic, and symmetric under exchange of ion species.The direct-correlation functions also obey pair-exchange symmetry.Source: S1. Properties that hold by construction · pp. 14–15
• The scalar free-energy form makes the one-body field integrable, so line integrals between density profiles give path-independent free-energy differences.Directly learned one-body correlations need not satisfy the Helmholtz condition exactly.Source: S1. Properties that hold by construction · p. 15
• The functional satisfies Noether sum rules for every specified density field, both in and out of equilibrium.Source: S1. Properties that hold by construction · p. 15
• For stable, noncritical homogeneous states with an invertible short-range response matrix and nonzero charge projection, the functional satisfies perfect-screening sum rules.The explicit Coulomb term fixes the leading k^2 charge-response coefficient; short-range correlations enter at order k^4.Source: S1. Properties that hold by construction · pp. 16–17

S2.1 Model and data

• The model and data section specifies polymer–ion interactions, equilibration at zero pressure, held-out and validation runs, and density and internal-force profiles recorded in 0.1σ bins.Source: S2.1 Model and data · p. 18

S2.2 Force matching

• Force matching uses the local force-balance relation, and the Fourier-weighted loss modestly improves long-wavelength amplitudes while leaving profile errors unchanged.The reported change in Γ(k1) remains within the molecular-dynamics uncertainty.Source: S2.2 Force matching · pp. 18–19

S2.3 Predictions and uncertainty

• Predictions use an accelerated Euler–Lagrange solver with a stated convergence threshold, and the main-result functionals have no negative eigenvalues of S(k)^−1 at sampled states and wavenumbers.The supplementary methods also define structure-factor estimates and their uncertainty from time blocks and independent runs.Source: S2.3 Predictions and uncertainty · p. 19

S3.1 Predictions across concentrations

• At strong coupling, the lowest concentration is the least accurate state for the number structure factor.The reported tables cover concentration-resolved force residuals and profile errors, plus bulk structure-factor errors and finite-wavenumber number response.Source: S3.1 Predictions across concentrations · p. 19, Table S2 · p. 20, Table S3 · p. 20
• At εr=7.5, the pair closure’s held-out residual rises from 0.70 at c=0.04 to 1.30, 1.55 and 2.19 as additional concentrations are included.At c=0.04, its cation/anion profile errors are 2.5/1.3%.Source: S3.1 Predictions across concentrations · p. 20

S3.2 Number response

• The number-response estimate neglects number–charge coupling, which vanishes as k→0 and contributes 1–4% of Γ at k1 in the reported models.The passage states that the MD estimate used in the main text is obtained by neglecting this correction.Source: S3.2 Number response · p. 20

S4.1 Larger boxes and nonplanar densities

• In boxes twice the training length, planar-trained models reproduce profiles under aperiodic potentials with 1.4–1.6% errors without a long mode and 0.8–1.6% with one, versus 2.4–6.2% for the pair closure.The long mode has wavenumber k1/2 and is included in two of the four tests.Source: S4.1 Larger boxes and nonplanar… · p. 21
• In two-dimensional aperiodic potentials, the neural functional’s profile errors are 1.8–3.1%, near MD noise of 1.7–3.0%, versus 3.6–5.5% for the pair closure and 31–62% for Poisson–Boltzmann theory.On periodic potentials, its errors are 0.6–3.7%, compared with 2.4–7.6% for the pair closure.Source: S4.1 Larger boxes and nonplanar… · p. 21, Table S5 · p. 21

S4.2 Original formulations, adaptations and comparison protocol

• The architecture comparison replaces excess free-energy terms or chemical potentials and trains the adaptations with the same force-matching protocol, data splits and model-selection procedure.The comparison tests adapted architectures under a common protocol, not published methods using their original training targets.Source: S4.2 Original formulations,… · p. 22
• Extending the lattice stencil from 1.5σ to 3σ has mixed effects: validation residuals rise sevenfold at εr=7.5 and 27-fold for the Lennard-Jones fluid, while εr=2 profile error falls from 7.2% to 5.2%.At εr=2, the Euler–Lagrange solution fails for 15 rather than 10 of 42 test runs with the larger radius.Source: S4.2 Original formulations,… · p. 22

S4.3 Data efficiency and structural consistency

• The section separately evaluates structural errors and data efficiency; its structural-identity residuals are distinct from predictive accuracy.The data-efficiency table reports mean held-out and untrained-concentration profile errors across training fractions, excluding nonconverged or density-spike solutions.Source: S4.3 Data efficiency and structural… · p. 23, Table S6 · p. 23, Table S7 · p. 23
• On finer and coarser grids, the neural functional’s mean chemical-potential changes are below 10^-9 and 3 × 10^-6, while the one-body network changes by 65–79% and the lattice free energy by 62–77%.The convolutional free energy cannot be evaluated on resampled grids because its read-out fixes the number of grid points.Source: S4.3 Data efficiency and structural… · p. 23

S4.4 Convolutional free-energy baseline

• Under force matching, the convolutional free energy has larger force residuals, few converged Euler–Lagrange solutions, and no uniquely defined bulk structure factor because its bulk Hessian is not diagonal in k.Its scalar output preserves integrability, but pooling and read-out break translation and reflection invariance; these findings characterize the architecture under force matching, not the published pair-correlation-trained method.Source: S4.4 Convolutional free-energy baseline · p. 23, Table S8 · p. 23