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Scalable long-range machine learning interatomic potentials with electrical response for atomistic material modeling

Yicheng Chen, Xiaoyun Yu, Hyejeong Song, Xintian Wang, Zihui Li, Zixi Xia, Junmin Chen, Peichen Zhong

arXiv:2610.23643v1cond-mat.mtrl-scics.LG

TL;DR

Device-scale atomistic modeling under electric fields needs long-range electrostatics and electrical response without losing the scalability of local MLIPs. DPA4C-LR separates global electrostatics from local latent-charge derivatives, enabling distributed evaluation across boundary conditions. It scales to millions of atoms and infers BECs from energy-and-force training, with a limited descriptor receptive field as a stated boundary.

  • Problem

    Atomistic simulations under electric fields need long-range electrostatics and electrical response at device-relevant scales, while strictly local MLIPs omit long-range interactions and field response.

    Source: §I · p. 1

  • Method

    DPA4C-LR analytically separates long-range electrostatic derivatives from local derivatives of learned latent charges, preserving spatial decomposition for distributed evaluation.

    Source: §I · p. 2

  • Results

    DPA4C-LR reaches 40% parallel efficiency up to 5.2 million atoms on four GPUs, while its long-range term costs 2.2 to 2.7 times a short-range DPA4C step on one GPU.

    Source: §IV · p. 10

  • Takeaways & Limitations

    Distributed evaluation of forces and BECs provides a practical foundation for investigating field-driven and interfacial dynamics at device-relevant scales.

    Source: §IV · p. 11

    The strictly local descriptor’s single-cutoff receptive field may limit its capture of medium-range interactions compared with message-passing architectures.

    Source: §IV · p. 10

Abstract

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Accurate atomistic simulations of materials under electric fields require long-range electrostatics and electrical-response modeling at length scales beyond the reach of conventional first-principles methods. Yet these capabilities remain difficult to reconcile with the scalability of machine-learning interatomic potentials (MLIPs). Strictly local models enable efficient spatial parallelization but neglect long-range interactions, whereas existing electrostatic extensions introduce global dependencies that hinder distributed computation. Here, we overcome this limitation by analytically separating the long-range electrostatic forces and Born effective charges (BECs) into a global electrostatics calculation and strictly local derivatives of learned latent charges. This formulation preserves the spatial decomposability of equivariant MLIPs while enabling distributed evaluation of energies, forces, stresses, and BECs across different boundary conditions. Implemented as DPA4C-LR on the \texttt{nvalchemi} platform, the framework scales to 5.2 million atoms on four GPUs at approximately twice the computational cost of its short-range backbone. Trained on energies and forces, DPA4C-LR accurately infers BECs and reproduces the field-dependent infrared spectra of water. We further demonstrate its capabilities by recovering the characteristic slim polarization hysteresis of relaxor \ce{Pb(Mg_{1/3}Nb_{2/3})O3}, which is obscured by finite-size effects in smaller simulations; predicting a room-temperature field-driven ionic conductivity of $6.8 \pm 0.7$~mS/cm in amorphous \ce{Li_{0.9}NbO_{0.9}Cl_{4.1}}; and modeling interfacial thermal transport across Si/diamond interfaces with slab-corrected electrostatics. By combining long-range physics, electrical-response inference, and distributed computation, this framework extends predictive atomistic modeling toward device-relevant length scales.

I. INTRODUCTION

Device-scale atomistic modeling under coupled thermal and electrical fields requires ab initio accuracy, long-range electrostatics with field response, and device-relevant length and time scales. Strictly local MLIPs scale efficiently but omit long-range interactions and electrical response, motivating a distributed long-range approach.

I. INTRODUCTION

  • Strictly local equivariant MLIPs support spatial partitioning and high throughput, but finite cutoffs cannot represent 1/r electrostatics or provide a variable for external fields to act on.Long-range interactions matter for charged defects, polar interfaces, and longitudinal–transverse optical phonon splitting.Source: §I · p. 1
  • Born effective charges connect atomic displacements and applied electric fields to cell-dipole and force responses, enabling infrared and dielectric response as well as finite-field simulations.Latent Ewald summation can infer polarization and BECs from charges learned using energies and forces alone, but its whole-cell computation impedes domain decomposition.Source: §I · p. 2
  • DPA4C-LR analytically separates long-range electrostatic derivatives from local latent-charge derivatives, preserving spatial decomposition while evaluating energies, forces, stresses, and BECs across boundary conditions.The implementation combines the DPA4C backbone with PME electrostatics and supports non-periodic systems, 3D periodic boundaries, and interface slabs with dipole correction.Source: §I · p. 2
  • 5.2 million atoms are reached on four GPUs, with 43% parallel efficiency at 1.96 million atoms and 2.1× long-range overhead over the short-range backbone.The work demonstrates applications to relaxor-ferroelectric hysteresis, ionic transport in amorphous LNOC, and thermal transport across silicon/diamond interfaces.Source: §I · p. 2

A. Model design

DPA4C-LR combines a strictly local equivariant representation with learned latent charges and a long-range electrostatics calculation. The design separates short- and long-range Coulomb contributions while supporting multiple boundary conditions.

A. Model design

  • DPA4C maps each atom’s local environment to a descriptor using an equivariant graph neural network with one message-passing layer, allowing independent evaluation across spatial partitions.The descriptor uses neighboring atoms within the cutoff radius, their atomic numbers, and positions.Source: A. Model design · p. 2
  • Separate fitting networks predict short-range atomic energies and raw latent charges, which are projected to enforce the system’s prescribed total charge.The projection guarantees that the charges sum exactly to the total system charge.Source: A. Model design · p. 2
  • Gaussian screening separates Coulomb interactions into a rapidly decaying short-range contribution absorbed into the short-range energy and a smooth long-range contribution evaluated from latent charges.The long-range energy is computed directly in real space for nonperiodic systems, with Ewald/PME for 3D periodic systems, and with a Yeh–Berkowitz correction for 2D slabs.Source: A. Model design · pp. 2–3
  • The workflow combines per-partition short-range energies and raw charges with a global Ewald/PME calculation that enforces total charge and evaluates long-range energy derivatives.Local automatic differentiation supplies the remaining position derivatives, which are assembled into total energies, forces, and BECs.Source: Figure 1 · p. 3

B. Distributed parallelization

The parallel implementation spatially partitions strictly local calculations while using global electrostatics for long-range terms. Forces and stresses are assembled from local and global derivative contributions across partitions.

B. Distributed parallelization

  • Each device handles a spatial partition containing owned atoms and ghost atoms that provide complete local environments within the model cutoff.DPA4C’s strict locality enables the short-range calculations to be spatially decomposed.Source: B. Distributed parallelization · p. 3
  • Short-range energies and raw latent charges are evaluated independently per partition, then projected charges feed the global long-range electrostatics calculation.This produces the total energy from partitioned short-range contributions and the long-range term.Source: B. Distributed parallelization · p. 3
  • Long-range forces combine direct electrostatic forces and charge-response terms; global charge gradients are paired with local automatic differentiation of latent-charge coordinates.Short-range and charge-response forces are summed by global atom indices across partitions, and stresses are evaluated analogously.Source: B. Distributed parallelization · p. 3

C. Born effective charges

DPA4C-LR obtains Born effective charges from screened latent charges using a direct-charge and charge-response decomposition. The charge-response derivatives are local and can be evaluated across spatial partitions.

C. Born effective charges

  • Latent charges are rescaled by the square root of the high-frequency relative permittivity to account for electronic screening before polarization and BEC evaluation.The BEC expression decomposes each tensor into a direct charge contribution and a charge-response contribution.Source: C. Born effective charges · p. 4
  • The pairwise response uses the identity matrix and the vector from atom i to atom j, with the outer product forming the tensor contribution.Here, rij denotes the displacement vector and ⊗ denotes the outer product.Source: C. Born effective charges · p. 4
  • The raw-charge derivatives in the pairwise BEC response are strictly local, so partitions can evaluate them independently and scatter-sum results by global atom indices.For configuration-independent screening, the global charge-projection term cancels from this response expression.Source: C. Born effective charges · p. 4

A. Bulk water

In bulk water, DPA4C-LR is trained on energies and forces alone, then evaluated for energy, force, and BEC accuracy, field-dependent infrared spectra, and multi-GPU scaling. It reproduces the reported water spectra and scales to 5.2 million atoms on four GPUs.

A. Bulk water

  • Force RMSE falls from 25.3 to 20.2 meV/˚A while energy RMSE remains comparable to DPA4C at 0.25 versus 0.22 meV/atom.The force RMSE is also below the 36.6 meV/˚A reported for both qNEP M1 and M2 variants.Source: Figure 2a
  • For the reported comparison, the resulting Born effective charges (BECs) are more accurate than those of qNEP, with an RMSE of 0.05 versus 0.10 e, respectively.Source: A. Bulk water · p. 4
  • DPA4C-LR reproduces bulk-water intramolecular, librational, and hydrogen-bond translational IR bands, including reported field-induced band shifts.The spectra are compared with experiment and cover static fields from 0 to 0.15 V/˚A.Source: A. Bulk water · p. 4
  • At 5.2 million atoms, DPA4C-LR runs at 493 ms/step on four GPUs, or 582 ms/step with on-the-fly BEC evaluation.At 1.96 million atoms, scaling from one to four GPUs yields speedups of 1.73× without BEC inference and 1.74× with it.Source: A. Bulk water · p. 4

B. Pb(Mg1/3Nb2/3)O3 relaxor ferroelectrics

PMN’s chemical ordering and relaxor response require large supercells to capture polarization hysteresis without finite-size effects. DPA4C-LR infers BECs from energy-and-force training and recovers the slim, nearly closed room-temperature loop in the largest tested cell.

B. Pb(Mg1/3Nb2/3)O3 relaxor ferroelectrics

  • PMN’s nanometer-scale chemical ordering and oxygen-octahedral distortions disrupt long-range polarization order, motivating large supercells for its P–E loop.The compared cells are 6 × 6 × 6, 12 × 12 × 12, and 18 × 18 × 18, with fields along [111] at 300 K.Source: B. Pb(Mg1/3Nb2/3)O3 relaxor… · pp. 5–6, Figure 3 · p. 6
  • Against 200 PBEsol DFT configurations, DPA4C-LR reaches energy and force RMSEs of 0.36 meV/atom and 40.3 meV/˚A, versus 0.43 meV/atom and 46.9 meV/˚A for DPA4C.The PMN model was distilled from the fine-tuned DPA-3 model.Source: B. Pb(Mg1/3Nb2/3)O3 relaxor… · p. 5
  • At 300 K along [111], the 18 × 18 × 18 PMN supercell reproduces a slim, nearly anhysteretic loop, whereas the 6 × 6 × 6 cell has a square loop with Pr = 22.4 µC/cm2.The loop shape is finite-size-sensitive; cells of tens of thousands of atoms are needed to recover the relaxor response.Source: Figure 3 · p. 6, B. Pb(Mg1/3Nb2/3)O3 relaxor… · pp. 6–7, Figure 3d
  • The ratio of clustered Nb sites rises from 0.53 to 0.67 within 4 × 10^4 swap iterations, while the largest cluster reaches 0.28 after 1.4 × 10^5 iterations.The 18 × 18 × 18 supercell accommodates chemically ordered Nb–Mg domains and Nb-rich domains.Source: B. Pb(Mg1/3Nb2/3)O3 relaxor… · pp. 5–6
  • In 30 PMN configurations, inferred diagonal BEC components correlate with DFPT at 0.97, and off-diagonal components have an RMSE of 0.22 e.The BECs were inferred by a model trained only on energies and forces.Source: B. Pb(Mg1/3Nb2/3)O3 relaxor… · p. 6

C. Ionic transport in an amorphous solid electrolyte

Field-driven molecular dynamics with on-the-fly Born effective charge inference estimates room-temperature ionic conductivity in amorphous LNOC. The section also reports model accuracy, large-cell simulations, and comparisons with zero-field and experimental estimates.

C. Ionic transport in an amorphous solid electrolyte

  • Training used energies and forces without charge or BEC supervision, with one melt and seven associated glasses held out from 29,257 labeled frames.The training trajectories came from a 662-atom cell and included crystalline, liquid, and glass configurations.Source: C. Ionic transport in an amorphous… · p. 7
  • DPA4C-LR reaches 0.65 meV/atom energy RMSE and 79 meV/Å force RMSE, versus 0.81 meV/atom and 100 meV/Å for short-range DPA4C on held-out frames.DPA4C-LR has lower energy and force errors in every configuration family.Source: C. Ionic transport in an amorphous… · p. 7
  • The four 19,872-atom glasses were quenched from one melt and simulated at 300 K under six static fields after 10 ps equilibration, with 1 ns trajectories.A glass MD step takes 8.8 ms on one H200 GPU.Source: C. Ionic transport in an amorphous… · p. 7
  • At 300 K, six fields from 0.0020 to 0.0064 V/Å yielded ionic conductivities of 6.1–7.6 mS/cm; their inverse-variance-weighted mean was 6.8 ± 0.7 mS/cm (95% confidence interval), consistent with linear response.Source: C. Ionic transport in an amorphous… · p. 7
  • The zero-field Einstein–Helfand estimate is 7 ± 5 mS/cm, with a confidence interval seven times wider than the field-driven estimate.The zero-field estimate uses thermal fluctuations of charge displacement; the field-driven estimate uses its field-direction drift.Source: C. Ionic transport in an amorphous… · p. 7
  • The field-driven value lies within the measured 1.7–9.4 mS/cm range for Li1–xNbO1–xCl4+x, but the comparison is qualitative because samples are partly amorphous and impedance spectra do not separate bulk from grain-boundary resistance.The measurements are reported at 25 °C for x from −0.10 to 0.20.Source: C. Ionic transport in an amorphous… · p. 8

D. Thermal transport at interfaces

DPA4C-LR predicts Si/diamond interfacial thermal conductance across 11–32 nm cells, with values near a prior simulation and slightly above the reported experimental range. The Yeh–Berkowitz correction brings slab electrostatics into close agreement with direct 2D Ewald summation.

D. Thermal transport at interfaces

  • The study uses NEMD with 330 and 270 K Langevin thermostats; heat flux comes from bath energy exchange and the interfacial temperature drop from extrapolated profiles.The Si–C potential was trained on crystal, amorphous, surface, and interface environments and fine-tuned on r2SCAN DFT data.Source: D. Thermal transport at interfaces · pp. 8–9
  • 0.148–0.160 GW m−2 K−1 is the DPA4C-LR interfacial thermal conductance across 11–32 nm cells, near the prior simulation’s about 0.16 GW m−2 K−1 and slightly above the experimental 0.064–0.14 GW m−2 K−1 range.The authors describe the predictions as consistent for this interface with the Yeh–Berkowitz correction implemented.Source: D. Thermal transport at interfaces · p. 9
  • At the 40.3 Å production vacuum gap, Yeh–Berkowitz-corrected 3D Ewald energy agrees with direct 2D Ewald within 2 × 10^-9 eV, while uncorrected 3D Ewald deviates by 0.88 eV.Uncorrected 3D Ewald retains spurious dipole interactions between periodic images even at an 80 Å gap.Source: D. Thermal transport at interfaces · p. 9, Figure 5 · p. 9

IV. DISCUSSION

The discussion presents a distributed long-range MLIP with a cell-size-independent real-space BEC expression, and reports scaling to multi-million-atom systems. It also identifies limits from local descriptors and latent charges.

IV. DISCUSSION

  • LES's finite-wavevector BEC approaches the finite-system definition only as k → 0, and its finite-size error decreases in larger cells without necessarily underestimating the total BEC.The geometric correction is of order L^-2 when the local charge response remains unchanged.Source: §IV · p. 9
  • The real-space BEC expression is cell-size independent and applies to non-periodic systems, three-dimensional periodic cells, and slabs.Figure 6 compares diagonal Nb BECs from DPA4C-LR's Eq. (10) and CACE-LR with LES's Eq. (15), normalized to n = 12.Source: §IV · p. 10, Figure 6 · p. 10
  • Analytical differentiation separates the global electrostatic solve from local charge-network derivatives, preserving spatial partitioning for forces, stresses, and BECs.The electrostatics solver supplies one scalar per atom for each partition's local backward pass; BEC evaluation adds another local backward pass.Source: §IV · p. 10
  • At 5.2 million atoms on four GPUs, weak-scaling efficiency remains 40%, while BEC computation adds less than 20% to force evaluation.For a 1.96-million-atom cell, four-GPU speed-up is 1.73 with 43% strong-scaling efficiency.Source: §IV · p. 10
  • A single-cutoff descriptor may miss medium-range interactions, while local latent charges cannot fully capture nonlocal charge redistribution in metals or metal–electrolyte interfaces.The discussion identifies global charge equilibration as a general way to describe such redistribution, with added computational and communication costs.Source: §IV · p. 10
  • Hardware-aware GPU execution and FPGAs are proposed as future routes to accelerate massive-scale simulations under external fields.The discussion envisions device simulations with millions to billions of atoms.Source: §IV · p. 10
  • The framework provides a basis for field-driven and interfacial simulations, with reported applications grounded in ab initio data at 10^4–10^6 atoms.The discussion attributes these simulations' viability to inherited domain decomposition and the real-space BEC expression.Source: §IV · pp. 9, 11
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