Latent Ewald Summation in Machine Learning

• LES is a long-range interaction framework that replaces explicit physical charges with latent variables derived from local atomic descriptors.
• It couples short-range machine-learning potentials with a reciprocal-space Ewald summation to globally propagate local information across periodic systems.
• LES demonstrates improved energy, force, and dielectric responses in applications from molecular dimers to water–vapor interfaces, and aids in the distillation of compact ML models.

Searching arXiv for papers on Latent Ewald Summation and closely related formulations. Latent Ewald Summation (LES) is a long-range interaction framework in which a machine-learned local representation is coupled to an Ewald-type global term through per-atom latent variables. In its explicit machine-learning form, a short-range ML interatomic potential predicts a latent quantity for each atom from local descriptors, and an Ewald summation over those latent variables supplies a nonlocal energy contribution; the model is trained end-to-end using energies and forces, without direct supervision of charges, dipoles, or external fields (Cheng, 2024). Subsequent work used the same framework to distill electrostatics that are latent in foundation ML interatomic potentials, yielding compact student models with access to Born effective charge tensors and infrared spectra (Wang et al., 12 Jun 2026).

1. Definition and position within Ewald-type methods

LES is best understood relative to conventional Ewald summation. Standard Ewald methods split the Coulomb kernel into a short-ranged real-space part and a smooth reciprocal-space part, thereby converting a conditionally convergent lattice sum into two rapidly convergent series. LES retains the Ewald idea of a global periodic long-range coupling, but replaces explicit physical charges with latent variables predicted from local atomic environments (Cheng, 2024).

In the original machine-learning formulation, the central modeling move is not to infer a unique physical charge partitioning, but to introduce a learned latent field whose reciprocal-space Ewald energy complements a local short-range model. This differs from charge-learning schemes that supervise partial charges or Wannier centers, and from purely local message-passing models whose effective interaction range is limited by cutoff times depth. The original LES work argues that message passing can improve finite-range information propagation but cannot fully emulate true long-range dielectric response in periodic systems (Cheng, 2024).

A broader interpretation of LES emerges from related Ewald literature. Several non-ML constructions replace or compress the explicit long-range term through auxiliary densities, analytic corrections, or Gaussian latent modes. These works do not use the name LES, but they clarify what it means for the reciprocal-space contribution to be present only implicitly, or “latently,” in a reformulated electrostatics calculation (Pickard, 2018).

2. Core mathematical formulation

In the original LES construction, each atom $i$ carries a local descriptor $B_i$. Two learned heads are applied to that descriptor: a short-range energy head and a latent-variable head,

$$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$

The latent variables $q_i$ are then inserted into an Ewald-like reciprocal-space energy (Cheng, 2024).

The latent structure factor is

$$S(\mathbf{k}) = \sum_i q_i e^{i\mathbf{k}\cdot\mathbf{r}_i},$$

with reciprocal vectors

$$\mathbf{k} = \left( \frac{2\pi n_x}{L_x}, \frac{2\pi n_y}{L_y}, \frac{2\pi n_z}{L_z} \right),$$

for an orthorhombic cell of volume $V=L_xL_yL_z$. The long-range term is written as

$$E^{lr} = \frac{1}{V} \sum_{0<k<k_c} \frac{e^{-\sigma^2 k^2/2}}{k^2} |S(\mathbf{k})|^2,$$

and the total energy is

$E = E^{sr} + E^{lr}.$

Only the reciprocal-space term is retained. The standard real-space and self-interaction terms of classical Ewald are omitted, because the latent field is not required to correspond to a physical electrostatic potential (Cheng, 2024).

The formulation is differentiable throughout. Gradients of $E^{lr}$ propagate both through the explicit dependence on atomic positions in the phase factors and through the implicit dependence of $B_i$0 on the local descriptors. As a result, forces are obtained by differentiating the total energy with respect to atomic coordinates, and the Ewald layer can be trained end-to-end with the short-range model (Cheng, 2024).

The latent variable need not be scalar. The original paper states that $B_i$1 can be multi-dimensional, in which case the total long-range energy is aggregated over contributions from different dimensions of $B_i$2 after the Ewald summation. In the dimer, bulk water, and water–vapor interface examples, the reported LES models use a 4-dimensional $B_i$3 (Cheng, 2024). In the later distillation setting, all student models use the LES augmentation with a one-dimensional latent charge variable, $B_i$4 and $B_i$5 ($B_i$6) (Wang et al., 12 Jun 2026).

3. Interpretation of the latent variable

A recurring misconception is that LES is simply a partial-charge model. The original formulation explicitly rejects that identification. The latent quantity $B_i$7 is not required to be a physically meaningful charge, and the model imposes no charge neutrality constraint, no absolute charge magnitude constraints, and no explicit charge equilibration. The omission of the usual Ewald self-interaction term is part of the same design choice: the latent field is optimized only insofar as it helps the total model reproduce reference energies and forces (Cheng, 2024).

Two interpretations are nevertheless central. First, LES can be viewed as an effective environment-dependent charge model. If the underlying long-range physics is approximately representable by charges, the learned latent variables can play that role operationally. Second, LES can be viewed as a global message-passing mechanism: local information is encoded in $B_i$8, and reciprocal-space Ewald coupling allows that local information to interact across the entire periodic simulation cell (Cheng, 2024).

Later distillation work sharpened the physical interpretation without changing the basic latent logic. In that setting, scalar latent charges $B_i$9 are predicted from local environments, passed into an LES Ewald energy, and then used to construct polarization, Born effective charges, and infrared response. Those quantities are not supervised directly; they emerge because the latent charges and their environmental dependence have been trained to reproduce teacher or DFT energies and forces (Wang et al., 12 Jun 2026).

This suggests a useful distinction between “latent” and “observable.” The latent variables are auxiliary degrees of freedom defined by the model architecture and loss. Physical response quantities become available only after the latent field is combined with a fixed electrostatic decoder, namely the Ewald-based long-range term and the corresponding polarization formalism (Wang et al., 12 Jun 2026).

4. Demonstrated behavior in dimers, bulk water, and interfaces

The original LES paper motivates the method through failures of purely short-ranged ML interatomic potentials in systems dominated by long-range electrostatics. In charged and polar molecular dimers, bulk water, and a water–vapor interface, standard short-ranged MLIPs can lead to unphysical predictions even when employing message passing, whereas the long-range LES models effectively eliminate these artifacts, with only about twice the computational cost of short-range MLIPs (Cheng, 2024).

For molecular dimers, the reported test set comprises charged–charged, charged–polar, polar–polar, charged–apolar, polar–apolar, and apolar–apolar classes. The short-range model uses a 5 Å cutoff and one message-passing layer, giving an effective perceptive field of about 10 Å. On test separations up to 15 Å, short-range models fail to extrapolate the binding curves, with flattening of the interaction energy and large force outliers. For charged and strongly polar dimers, LES reduces energy and force RMSEs by about an order of magnitude and tracks the reference DFT potential across the separation range (Cheng, 2024).

For bulk liquid water, the key distinction is between structural and dielectric observables. The O–O radial distribution function is reproduced well by both short-range and LES models, confirming that short-range MLIPs can be adequate for local structure. By contrast, the long-wavelength dipole–dipole correlation function diverges as $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$0 for short-range models. Message passing shifts the pathology to somewhat longer length scales but does not remove it. LES suppresses this divergence and restores physically reasonable small-$$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$1 behavior (Cheng, 2024).

For the water–vapor interface, all models reproduce similar density profiles, but the orientational order reveals the missing long-range screening in short-range models. In the short-range $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$2 model, the interface dipole layer is not properly screened and orientational order persists deep into the bulk. In the short-range $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$3 model, message passing reduces the artifact in thinner slabs but, for thicker slabs, produces dipole ordering of the opposite sign in the bulk, and some trajectories become unstable. The LES models reproduce the DFT reference behavior: the interface dipole layer is represented, screening occurs within a few layers, and the bulk has negligible net dipole (Cheng, 2024).

These results are often taken as the principal empirical argument for LES. The method is not introduced to improve short-range structural metrics that are already captured well by local models; it is introduced to restore the missing nonlocal electrostatics and dielectric response that remain inaccessible to finite-range architectures (Cheng, 2024).

5. Distillation, electrical response, and spectroscopy

A later development uses LES as a distillation mechanism for foundation ML interatomic potentials. Energies and forces predicted by a teacher model are used to train a lightweight LES-augmented student MLIP, with optional fine-tuning on additional DFT data. The resulting models reduce computational cost while providing access to Born effective charge tensors and infrared spectra, and the benchmark spans students distilled from UMA, MACE, Orb, eSEN, GemNet-OC, PET, and EquiformerV2-based models (Wang et al., 12 Jun 2026).

In this framework, LES is a modular global layer attached to a short-range baseline. Two student architectures are emphasized: MACELES for molecular liquids and CACELES for $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$4–water interfaces. The reported MACELES students have $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$5 M parameters, while CACELES models have $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$6 M parameters depending on whether message passing is enabled. The distillation paper states that these students are three to four orders of magnitude smaller than some teacher models (Wang et al., 12 Jun 2026).

The electrical-response formalism is built directly on the latent charge field. For periodic systems, LES uses a Fourier-space representation of the polarization, and the Born effective charge tensor of atom $$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$7 is written as

$$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$8

Infrared spectra are then obtained from the polarization current

$$E^{sr} = \sum_i E_\theta(B_i), \qquad q_i = Q_\phi(B_i).$$9

with

$q_i$0

No explicit dipole labels or electric fields are used in training; the spectra are a post-processing output of LES-MLIP trajectories (Wang et al., 12 Jun 2026).

The reported numerical behavior is specific. In liquid water, most distilled MACELES students achieve BEC $q_i$1 versus DFT, with RMSEs around 0.05 e when trained on as few as 25 configurations. The paper argues that BECs are learned very data-efficiently, and that teachers with too small a receptive field yield much poorer BEC accuracy even when energy and force reproduction remains acceptable (Wang et al., 12 Jun 2026).

The spectroscopic conclusions are equally pointed. Across water, concentrated hydrochloric acid, and the anatase $q_i$2-water interface, electrostatic response can be extracted from most foundation MLIPs, but the underlying DFT level and dataset used to train the teacher model play a larger role than architecture in determining electrostatic and spectroscopic accuracy. For the interface system, fine-tuning a distilled student with as little as 10% of the SCAN DFT forces, approximately 300 snapshots, yields water density and infrared spectra essentially indistinguishable from SCAN-level CACELES and DPLR references; increasing to 50% brings only marginal improvement (Wang et al., 12 Jun 2026).

6. Conceptual lineage and related latent Ewald constructions

LES in the strict sense refers to the machine-learning framework above, but several related Ewald constructions illuminate its mathematical content. They suggest that “latent Ewald” is not limited to latent atomic charges; it can also mean representing the long-range contribution implicitly through auxiliary continuous fields, corrected real-space formulas, or compressed spectral bases.

Pickard’s purely real-space method for ion–ion Coulomb energies in a uniform neutralizing background is a particularly close antecedent. It replaces the reciprocal-space part with a rigorously corrected real-space scheme based on a damped pair sum plus an analytic correction,

$q_i$3

and attains linear scaling for fixed cutoff (Pickard, 2018). The paper explicitly describes the approach as closely related to Wolf’s method for neutral ionic crystals. A plausible LES interpretation is that the missing reciprocal-space physics is carried by a local analytic correction rather than an explicit $q_i$4-space sum (Pickard, 2018).

The “model density approach to Ewald summations” provides a different kind of latent decomposition. A model density $q_i$5 is chosen so that the residual $q_i$6 has vanishing multipoles up to a chosen order. The potential of the full density is then expressed as an analytic model contribution plus a rapidly convergent Ewald sum for the residual (Ribaldone et al., 29 Jan 2026). This suggests a multipole-based LES viewpoint in which the low-order multipole coefficients act as a compact latent representation of the long-range field and the residual Ewald sum becomes a small correction (Ribaldone et al., 29 Jan 2026).

The $q_i$7-series offers yet another perspective. It constructs a separable Gaussian decomposition of the Coulomb kernel in which the near part is exact up to a cutoff radius and the far part is a sum of Gaussians with geometric scaling. The paper reports that the method is more accurate than the standard Ewald decomposition for a given amount of computational effort, and achieves the same accuracy as the Ewald decomposition with approximately half the computational effort (Predescu et al., 2019). In LES terms, this is a Gaussian latent-mode expansion of the kernel itself rather than of the charge density.

Taken together, these works define a broader conceptual field around LES. The explicit ML version learns latent atomic variables and inserts them into an Ewald layer (Cheng, 2024). The real-space background-corrected approach makes the reciprocal contribution latent in an analytic local correction (Pickard, 2018). The model-density approach makes it latent in auxiliary multipole-matched densities (Ribaldone et al., 29 Jan 2026). The $q_i$8-series makes it latent in a separable Gaussian mode expansion (Predescu et al., 2019). What unifies them is not a single algorithmic recipe, but a common strategy: preserve the boundary conditions and asymptotics of Ewald summation while relocating the expensive or delicate long-range content into a more compact, implicit, or structured representation.