Atomic Physics Neural Networks

• Atomic Physics Neural Networks (APNN) are methods that incorporate atomic-scale physics and symmetry constraints directly into neural architectures.
• APNN models use locality, invariant representations, and hierarchical aggregation to accurately predict potentials, wave functions, and other observables.
• They enable efficient configuration interaction, robust quantum control, and high-fidelity atomic-scale imaging by embedding physical rules into the learning process.

Atomic Physics Neural Networks (APNN) denotes a family of neural-network methods in which atomic-scale physics is embedded directly into the representation, architecture, training objective, or workflow. In the literature, the term spans several distinct but related uses: covariant networks for atomic potential energy surfaces, atomic neural networks for molecules and materials, neural quantum states for electronic and nuclear many-body problems, neural classifiers embedded in atomic configuration-interaction pipelines, physics-informed neural networks for atomic and optical quantum control, and physics-aware networks for atomic-scale structure determination from microscopy. Across these settings, the defining motif is not a single architecture but the explicit incorporation of physical structure such as local environments, symmetry, conservation laws, projection models, or Hamiltonian dynamics (Kondor, 2018, Shao et al., 2019, Bilous et al., 2024, Zhang et al., 20 Mar 2026).

1. Scope and defining characteristics

Within atomistic modeling, APNN usually refers to models that operate on atoms as fundamental units and construct local, symmetry-respecting representations of each atom’s environment before aggregating atomic contributions into extensive observables such as total energy, forces, virial stress, or other physico-chemical quantities. A standard formulation is the atomic-energy ansatz

$$E(\mathbf{R}) = \sum_i E_i\big(G_i(\mathbf{R})\big), \qquad \mathbf{F}_i=-\nabla_{\mathbf{R}_i}E(\mathbf{R}),$$

with locality enforced by finite cutoffs and scalar invariance obtained from internal coordinates or symmetry-adapted features (Shao et al., 2019). In a more representation-theoretic formulation, each subsystem carries a covariant state $\psi_j$ in a representation space of $SO(3)$, and parent states are built by aggregation rules of the form

$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$

with irreducible $SO(3)$ structure preserved by Clebsch–Gordan reduction and same-$\ell$ blockwise mixing (Kondor, 2018).

The label also extends beyond interatomic potentials. In atomic CI, APNN describes neural selectors that classify relativistic configurations by importance inside iterative pCI workflows, thereby reducing otherwise intractable configuration spaces (Bilous et al., 2024, Bilous et al., 3 Mar 2025). In atomic electron tomography, APNN describes physics-aware networks that encode projection geometry, HAADF contrast, local symmetry, and coordination-shell structure to recover atomic coordinates and species under low-dose conditions (Zhang et al., 20 Mar 2026). In quantum control, the term is used for PINN-based control of open quantum systems, where the GKSL or Schrödinger dynamics are embedded into the loss so that learned controls remain tied to atomic or optical Hamiltonians (Norambuena et al., 2022).

Family	Core task	Representative papers
Covariant atomistic networks	Atomic potentials and many-body energy mapping	(Kondor, 2018, Yoo et al., 2019)
Atomic neural-network libraries	PES, forces, stress, dipoles, charges	(Shao et al., 2019, Tam et al., 2019)
Neural wave functions	Excited states, ionization, nuclear ground states	(Liu et al., 2023, Gnech et al., 2021)
CI-integrated APNN	Selection of important relativistic configurations	(Bilous et al., 2024, Bilous et al., 3 Mar 2025)
Physics-informed control and imaging	Quantum control and atomic tomography	(Norambuena et al., 2022, Zhang et al., 20 Mar 2026)
Nuclear-observable regression	Charge radii and atomic masses	(Akkoyun et al., 2012, Huang et al., 2 Jan 2025)

A common misconception is that APNN denotes a single standardized model class. The cited literature instead uses the term as an umbrella over several architectures whose shared feature is the direct encoding of atomic-scale physics into learning.

2. Symmetry-structured interatomic potentials and atomic energy mapping

A central APNN lineage concerns the learning of interatomic potentials and potential energy surfaces. N-body Networks formulate this problem as a compositional directed acyclic graph in which each node corresponds to a physical subsystem, carries a position $r_j\in\mathbb{R}^3$ and a covariant state $\psi_j$, and aggregates child subsystems through local tensorial rules. Rotational covariance is exact: activations transform under block-diagonal direct sums of Wigner $D$ matrices, nonlinearities are implemented by tensor products followed by Clebsch–Gordan decomposition, and learnable weights are permitted only within fixed-$\ell$ irreducible blocks. The resulting architecture is hierarchical, Fourier-space, and explicitly $\psi_j$0-body in structure rather than descriptor-based in the usual sense (Kondor, 2018).

PiNN and its PiNet architecture represent a more application-oriented APNN framework for molecules and materials. PiNet defines pairwise interaction features

$\psi_j$1

with radial basis expansions filtered by a smooth Behler cutoff,

$\psi_j$2

and zero biases in the II-layer to guarantee smooth PES behavior at the cutoff. PiNN implements both PiNet and Behler–Parrinello HDNNs, provides analytical virial pressure under periodic boundary conditions, and interfaces with ASE and a development version of AMS. Reported results include MAE $\psi_j$3 eV on QM9, MAE $\psi_j$4 eV/atom on Materials Project crystals, and MAE $\psi_j$5 eV/atom on a perovskite benchmark (Shao et al., 2019).

The APNN literature also distinguishes between accurate total-energy fitting and correct atomic-energy inference. In the atomic-energy mapping analysis for Si, the DFT total energy is written as

$\psi_j$6

and the neural approximation as

$\psi_j$7

The work emphasizes invariant points in feature space, where all atoms share the same local descriptor and the atomic energy is fixed by $\psi_j$8. It shows that networks can achieve energy RMSE $\psi_j$9 meV/atom and force RMSE $SO(3)$0 eV/Å on crystalline Si while still learning an incorrect EOS away from equilibrium, and that in a Si(100)-(2×2) slab at 100 K the bulk and surface mapping errors can be $SO(3)$1 meV/atom and $SO(3)$2 meV/atom despite a total-energy RMSE of $SO(3)$3 meV/atom (Yoo et al., 2019). This established a technical caution: APNN potentials may exhibit ad hoc mapping unless training sets span volumes, structures, and feature-space connectivity.

A deliberately simple dense-network variant was demonstrated for Al by fitting per-atom energies from EAM and MEAM using the spherical-coordinate input

$SO(3)$4

with a nine-layer fully connected network. More than $SO(3)$5 of EAM samples and more than $SO(3)$6 of MEAM samples had relative error below $SO(3)$7; over $SO(3)$8 of EAM samples and over $SO(3)$9 of MEAM samples had absolute error below $$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$0 meV (Tam et al., 2019). This suggests that even non-invariant representations can approximate classical reference potentials accurately on restricted tasks, although the broader APNN literature strongly favors explicit invariance and smoothness.

3. Neural wave functions and many-body structure in atomic and nuclear physics

A second APNN lineage uses neural networks as variational many-body ansätze. In the excited-state framework based on FermiNet with effective core potentials, the many-electron Hamiltonian is written as

$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$1

with semi-local ECP operators

$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$2

The neural wave function has the FermiNet-like form

$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$3

and excited states are targeted sequentially with an orthogonality-penalized loss

$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$4

The reported vertical excitation energies for atoms from Li to Br and molecules including LiH, BeH, CO, H$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$5O, H$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$6S, H$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$7CSi, and benzene have mean absolute errors typically in the $$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$8–$$\psi_j=\Phi_j(\hat r_{ch_1},\ldots,\hat r_{ch_k},\psi_{ch_1},\ldots,\psi_{ch_k}),$$9 mHa range, with almost all level errors within $SO(3)$0 mHa versus theoretical best estimates or experiment (Liu et al., 2023).

Neural quantum states also appear in nuclear APNN. For nuclei up to $SO(3)$1, the variational ansatz is

$SO(3)$2

where $SO(3)$3 is an antisymmetrized mean-field state with good $SO(3)$4, while $SO(3)$5 and $SO(3)$6 are permutation-invariant neural correlators of Deep Sets form

$SO(3)$7

Benchmarks against hyperspherical harmonics show close agreement for $SO(3)$8H, $SO(3)$9H, $\ell$0He, $\ell$1He, $\ell$2He, and $\ell$3Li. For example, for $\ell$4Li with the NN-only Hamiltonian the ANN energy is $\ell$5 MeV versus $\ell$6 MeV from HH, while with NN+3N it is $\ell$7 MeV versus $\ell$8 MeV (Gnech et al., 2021). The underbinding relative to HH was attributed to nodal limitations of the mean-field part.

At the level of learned nuclear observables, a small ANN trained on around $\ell$9 nuclei yielded the empirical charge-radius formula

$r_j\in\mathbb{R}^3$0

with root-mean-square deviation $r_j\in\mathbb{R}^3$1 fm from experiment (Akkoyun et al., 2012). A physics-informed fully connected neural network for atomic masses adopted a macroscopic–microscopic decomposition with outputs $r_j\in\mathbb{R}^3$2 and reached a test RMSD of $r_j\in\mathbb{R}^3$3 MeV on AME2020 and $r_j\in\mathbb{R}^3$4 MeV when extrapolating from AME2016 to newly added AME2020 nuclei (Huang et al., 2 Jan 2025). Together, these works indicate that APNN in the many-body context ranges from direct wave-function representation to surrogate learning of derived observables.

4. APNN inside atomic configuration interaction and high-precision codes

A distinct atomic-physics usage of APNN is as a classifier embedded in the configuration-interaction workflow. In this setting the CI expansion

$r_j\in\mathbb{R}^3$5

is not replaced by a network; rather, a network learns which relativistic configurations should be retained before the CI Hamiltonian is built and diagonalized. Configuration importance is defined by

$r_j\in\mathbb{R}^3$6

and the network performs binary classification according to whether $r_j\in\mathbb{R}^3$7 for a threshold $r_j\in\mathbb{R}^3$8 that is lowered iteratively (Bilous et al., 2024).

The APNN-pCI workflow of Bilous, Cheung, and Safronova uses binary encodings of relativistic subshell occupancies as features and a dense MLP with four ReLU hidden layers and a softmax output. Labels come from previous pCI diagonalizations, while a balancing set of randomly added configurations prevents the classifier from collapsing onto already discovered subspaces. For Ni$r_j\in\mathbb{R}^3$9, the full pool of $\psi_j$0 relativistic configurations was reduced to a final APNN CI set of $\psi_j$1 configurations, a $\psi_j$2 reduction; the total pCI wall time fell from $\psi_j$3 h to $\psi_j$4 h, with final deviations from large direct-CI references of $\psi_j$5 and $\psi_j$6 cm$\psi_j$7 across the five lowest levels (Bilous et al., 2024). For Fe$\psi_j$8, reported RMS deviations are $\psi_j$9 cm$D$0 for five even-parity levels and $D$1 cm$D$2 for seventeen odd-parity levels.

The later Python package formalizes this methodology as a “Neural manager” on top of pCI and other high-performance atomic codes. It treats the APNN component as code-agnostic and delegates format-specific operations to an adapter layer. In the Fe$D$3 demonstration, the full even-parity CI basis contained $D$4 relativistic configurations, the prior run used $D$5, the APNN loop started from $D$6, and five iterations produced $D$7, $D$8, $D$9, $\ell$0, and $\ell$1 configurations. Timings were $\ell$2 min for full CI, $\ell$3 min for the prior run, $\ell$4 min for the start, and $\ell$5, $\ell$6, $\ell$7, $\ell$8, and $\ell$9 min for the five APNN iterations, giving $\psi_j$00 min total for the APNN route in this medium-size example (Bilous et al., 3 Mar 2025). The authors note that the benefit grows when full CI is impossible rather than merely expensive.

This CI-centered usage of APNN differs conceptually from neural potentials or neural wave functions. The network does not approximate the Hamiltonian or the eigenstate directly; it approximates the combinatorial selection rule that would otherwise require prohibitive perturbative screening or brute-force diagonalization.

5. Physics-informed control and atomic-scale measurement

APNN also includes models that encode the governing dynamics or measurement operator directly into the loss. In open-system quantum control, Norambuena et al. formulate the dynamics through the GKSL master equation

$\psi_j$01

or through the Schrödinger equation for closed systems. Their PINN outputs state and control trajectories using a hard initial-condition constraint

$\psi_j$02

and minimizes a loss containing physics residuals, fidelity terms, control penalties, and constraint penalties (Norambuena et al., 2022). In a two-level open system with damping, the PINN found a time-dependent Stark-like control $\psi_j$03 achieving Uhlmann fidelity $\psi_j$04 for $\psi_j$05, with each density-matrix component within $\psi_j$06 of the target steady state. In a three-level $\psi_j$07-system, the baseline result was $\psi_j$08, $\psi_j$09, and $\psi_j$10, compared with STIRAP at $\psi_j$11, $\psi_j$12, $\psi_j$13 (Norambuena et al., 2022). The work explicitly treats these PINNs as APNNs when applied to TLS, $\psi_j$14-systems, and NV-center-like platforms.

On the measurement side, the physics-aware, two-stage neural networks for atomic electron tomography define a forward model

$\psi_j$15

with low-dose Poisson noise, additive Gaussian noise, correlated backgrounds, and missing-wedge distortions. Stage 1, GLARE, is a global–local 3D ResUNet with FiLM modulation and an $\psi_j$16 regression loss against physically constructed positive target volumes. Stage 2, DAST, computes $\psi_j$17 3D Zernike coefficients per atom, concatenates them with normalized coordinates, builds a $\psi_j$18-NN graph with $\psi_j$19, and applies neighbor-restricted graph attention for elemental classification (Zhang et al., 20 Mar 2026).

The quantitative results are among the clearest demonstrations of APNN in atomic-scale imaging. On a benchmark of $\psi_j$20 reconstructed volumes, the typical-noise test gave median F1 $\psi_j$21 and median RMSD $\psi_j$22 Å for GLARE, compared with $\psi_j$23 and $\psi_j$24 Å for direct tracing. Overall under typical conditions, the full PANN pipeline reduced atomic coordinate error from $\psi_j$25 Å to $\psi_j$26 Å and raised the combined atomic recovery rate from $\psi_j$27 to $\psi_j$28. Full DAST achieved $\psi_j$29 mean classification accuracy, whereas coordinate-only DAST plateaued at $\psi_j$30 validation/test accuracy. On low-dose experimental data at $\psi_j$31 dose, two independent PANN reconstructions agreed at $\psi_j$32, versus $\psi_j$33 for conventional processing; for CsPbBr$\psi_j$34 at $\psi_j$35 e·Å$\psi_j$36, fine-tuned GLARE-CPB achieved F1 $\psi_j$37 and RMSD $\psi_j$38 Å (Zhang et al., 20 Mar 2026).

These examples show two complementary APNN modes: one constrains control synthesis by dynamical equations, and the other constrains reconstruction by the imaging operator, local symmetry descriptors, and coordination-aware graph structure.

6. Methodological principles, limitations, and terminological ambiguity

Across the literature, several design principles recur. One is explicit symmetry handling: translational and rotational invariance, permutation invariance, or full rotational covariance are built into descriptors, message passing, or irreducible representations rather than left to data augmentation alone (Kondor, 2018, Shao et al., 2019). A second is physically structured decomposition: atomic energies are summed over local environments, nuclear masses are split into $\psi_j$39 and residual terms, CI selection is performed at the level of relativistic configurations rather than individual determinants, and tomography separates global distortion correction from local symmetry-based species classification (Huang et al., 2 Jan 2025, Bilous et al., 3 Mar 2025, Zhang et al., 20 Mar 2026). A third is structure-preserving regularization: smooth cutoff functions, hard initial-condition constraints, trace or positivity penalties, and local-neighborhood attention are used to keep learned objects physically admissible (Shao et al., 2019, Norambuena et al., 2022).

The main limitations are equally consistent. In interatomic potentials, accurate total-energy fitting can conceal incorrect atomic-energy mapping, so invariant-point diagnostics and training-set connectivity matter (Yoo et al., 2019). In quantum control of open systems, exact GKSL satisfaction guarantees complete positivity and trace preservation only in the continuous limit; in practice, numerical residuals may drift unless Cholesky-like or exponential-map parameterizations are added (Norambuena et al., 2022). In atomic electron tomography, the workflow assumes no atomic dynamics during tilt acquisition and remains modular rather than end-to-end from raw tilt series to atomic model (Zhang et al., 20 Mar 2026). In nuclear mass and radius prediction, extrapolation remains limited by sparse data near drip lines and by the lack of calibrated uncertainty quantification (Akkoyun et al., 2012, Huang et al., 2 Jan 2025). In CI selection, performance depends on the quality of the prior CI seed and on the schedule of selection thresholds (Bilous et al., 2024, Bilous et al., 3 Mar 2025).

A further source of confusion is nomenclature. The acronym APNN is not unique across arXiv usage. In radiative-transfer literature it also denotes “Asymptotic-Preserving Neural Networks,” where the emphasis is multiscale kinetic-to-diffusion limits rather than atomic-scale structure or atomic physics (Wu et al., 14 Jan 2025). In the atomic-physics sense, however, the cited works converge on a coherent idea: neural networks become APNNs when they do not merely fit atomic data, but instead encode the governing symmetries, operators, or hierarchical structures of atomic-scale physics itself.

A plausible implication is that APNN should be understood less as a narrow subfield than as a design doctrine. Whether the task is a potential-energy surface, an excited-state wave function, a CI basis, a control field, or a tomography reconstruction, the defining move is the same: replace generic function approximation by a neural architecture whose hypothesis space has already been shaped by the physics of atoms.