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Recovering Irreducible Three-Body Interactions with Lagrangian Neural Networks

Sena Kalabalik, Sukru Caglar, Abdullah Umut Hamzaogullari, Arkadas Ozakin

arXiv:2610.13292v1cs.LGphysics.atom-phphysics.comp-ph

TL;DR

Prior Lagrangian neural-network studies have not addressed irreducible many-body interactions that cannot be represented by pairwise sums. This paper learns the Axilrod–Teller potential from trajectory-derived accelerations using domain-specific structure and signal-aware sampling. It reports recovery when the three-body signal exceeds model error, and symbolic regression recovers the interaction’s analytic structure and amplitude.

  • Problem

    Whether Lagrangian neural networks can recover irreducible many-body interactions from motion alone remains untested, despite their inability to be represented as sums of pairwise terms.

    Source: §I · p. 1

  • Method

    The approach combines domain-specific structure with sampling concentrated in informative regions to learn the Axilrod–Teller interaction from dynamics and the trained potential.

    Source: §VI · p. 12, §VII · p. 13

  • Results

    The three-body contribution is recovered when sufficiently large relative to residual model error, while symbolic regression recovers the Axilrod–Teller analytic structure and amplitude.

    Source: §VII · p. 13

  • Takeaways & Limitations

    The results demonstrate recovery of an irreducible three-body interaction from motion data and its conversion into an explicit physical law.

    Source: §VII · p. 13

    The Axilrod–Teller expression is an asymptotic long-range approximation, and the sampled configurations do not all lie in its regime of quantitative validity for real rare-gas atoms.

    Source: §III · p. 4

Abstract

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Lagrangian Neural Networks (LNNs) learn mechanical systems from trajectory data by parameterizing a scalar Lagrangian and deriving the dynamics through the Euler-Lagrange equations, biasing the model toward motion that respects the variational structure of mechanics. The interactions studied so far have been pairwise-additive, leaving open whether LNNs can recover an irreducible many-body interaction from motion alone. We address this using the Axilrod-Teller triple-dipole interaction, the leading three-body dispersion term, whose dependence on the collective geometry of three particles cannot be represented by any sum of pairwise terms. We prescribe the kinetic energy exactly and learn only the potential, as a function of interparticle distances and angles. Combined with a signal-aware sampling strategy that concentrates training data where the three-body signal is strongest while preserving broad coverage of configuration space, this recovers the interaction from accelerations alone. The learned potential reproduces the radial scaling, angular dependence, and characteristic sign reversal of the Axilrod-Teller interaction, and generates stable trajectories. Embedding it within a pairwise Lennard-Jones potential lets us examine its observability: the three-body term is recovered once its dynamical signature exceeds the residual modeling error, which it does cleanly at amplified coupling but not at the coupling of a real noble gas, bracketing the transition between the two. Finally, symbolic regression on the trained potential, with no assumed functional form, returns the Axilrod-Teller expression with angular numerator, radial law, and coupling constant intact. An LNN with domain-specific structure can thus recover an irreducible three-body interaction from trajectories as a closed-form law, pointing toward learning interaction laws whose structure is not known a priori.

I. INTRODUCTION

Lagrangian neural networks infer dynamics by differentiating a learned Lagrangian, but their study has not addressed irreducible many-body interactions. This paper uses the Axilrod–Teller interaction to test whether such structure can be recovered from trajectories and converted into an analytic law.

A. Lagrangian neural networks

  • Lagrangian neural networks parameterize a scalar Lagrangian and derive dynamics through the Euler–Lagrange equations, biasing predictions toward variationally consistent mechanics.Training compares accelerations derived from the learned Lagrangian with reference accelerations.Source: §I · p. 1, A. Lagrangian neural networks · p. 2

I. INTRODUCTION

  • The Axilrod–Teller interaction is an irreducible three-body target: its collective correlations cannot be reproduced by any composition of pairwise fits.It is a leading three-body correction to pairwise potentials, with a shape-dependent angular factor.Source: §I · p. 1
  • Symbolic regression is used to turn the learned potential into a candidate analytic expression, complementing tests of numerical agreement with trajectories.The paper frames compact analytic recovery as a way to identify what interaction the model learned.Source: §I · pp. 1–2

B. Learning Lagrangian and Hamiltonian dynamics

  • Prior Lagrangian and Hamiltonian learning includes nonlinear and chaotic systems, but the cited interaction models do not explicitly represent irreducible three-body terms.Graph-network approaches use edge-based or pairwise decompositions, leaving contributions such as Axilrod–Teller unaddressed.Source: B. Learning Lagrangian and Hamiltonian… · p. 2

C. The Axilrod–Teller interaction

  • The Axilrod–Teller potential scales as λ^-9 under uniform size rescaling, while its angular factor changes sign at cos θ1 cos θ2 cos θ3 = -1/3.For positive coupling, compact near-equilateral geometries are repulsive and elongated near-collinear geometries are attractive.Source: C. The Axilrod–Teller interaction · p. 2

D. Machine-learned interatomic potentials

Many learned interatomic potentials use quantum-mechanical energies or forces, whereas this work targets a specific interaction from classical trajectories without direct potential labels. The study uses reduced-unit three-particle models and limits its physical claims because the Axilrod–Teller form is an asymptotic approximation.

D. Machine-learned interatomic potentials

  • Unlike potentials trained on quantum-mechanical energetics, this study isolates one interaction and recovers it implicitly from classical trajectories through acceleration matching, without energy or potential labels.The distance-plus-angle-cosine representation enriches pairwise distances without engineered symmetry-function expansions.Source: D. Machine-learned interatomic… · p. 3

E. The Axilrod–Teller interaction in simulation

  • The Axilrod–Teller term has been used in rare-gas simulations, but it is an asymptotic long-range approximation whose assumptions fail when charge overlap and exchange effects become significant.Its quantitative reliability is expected only when atoms are sufficiently separated for charge overlap to be negligible.Source: E. The Axilrod–Teller interaction in… · p. 3, §III · p. 4

III. PHYSICAL SYSTEM DEFINITION

  • The study models three identical particles in two dimensions using reduced Lennard–Jones units, and treats its isolated and composite systems as tests of learnability rather than realistic material descriptions.Much of the sampled configuration space lies outside the regime where Axilrod–Teller quantitatively describes real rare-gas atoms.Source: §III · pp. 3–4

IV. METHODS

The method fixes known physical structure in the Lagrangian and learns the interaction potential from acceleration data. Training data are sampled to emphasize informative, stable regions while covering a broad configuration space.

IV. METHODS

  • The partially parameterized Lagrangian prescribes the kinetic term and learns only the interaction potential, training by matching accelerations derived through the Euler–Lagrange equations.The potential receives no direct supervision; its training signal comes through the accelerations it generates.Source: §IV · p. 4, Figure 2 · p. 4
  • The section identifies fixed kinetic structure and augmented input reparameterization as its two structural choices, with the network trained using mean absolute acceleration error.Table I is identified as situating the parameterization among existing Lagrangian networks.Source: §IV · p. 4, Table I · p. 4
  • Training and validation configurations are sampled from a log-uniform mixture of dense and sparse regions to emphasize the narrow regime where the interaction is informative and numerically stable.The sampling is described as an adaptive allocation of data to where the learning signal lives.Source: §IV · pp. 4–5

A. Fixed kinetic term

The architecture prescribes the known kinetic energy and concentrates learning on the unknown potential. It represents three-body geometry with distances and angle cosines, retaining Cartesian dynamics while exposing radial and angular structure to the network.

A. Fixed kinetic term

  • With the kinetic energy prescribed, the learning problem reduces to recovering a scalar potential from its gradient field, well-posed up to an additive constant where configuration space is covered.This concentrates network capacity on unknown interaction physics without reducing expressive power within the stated class of systems.Source: A. Fixed kinetic term · p. 5

B. Input reparameterization

  • The network receives three pairwise distances and three angle cosines, an augmented representation that exposes radial and angular structure without adding geometric information.The same representation recovers radial potentials in Lennard–Jones and Newtonian controls without spurious angular dependence, and also works for the composite LJ+AT system.Source: B. Input reparameterization · p. 5
  • Velocities do not enter the learned potential; differentiating it through the internal-coordinate features still yields dynamics in Cartesian coordinates.Both learned and reference accelerations are obtained by differentiating their respective potentials with respect to Cartesian positions.Source: B. Input reparameterization · p. 5

V. EXPERIMENTS

Experiments test whether an isolated Axilrod–Teller term can be learned and then assess recovery in pairwise-plus-three-body settings through field, geometric, and rollout evaluations. Dense/sparse sampling improves the equilateral-sweep acceleration fit, while the learned pure-AT potential and field closely match analytic results.

V. EXPERIMENTS

  • The experiments first isolate the AT interaction, then combine it with a dominant Lennard-Jones potential and vary the three-body strength to test identifiability.The isolated study evaluates field accuracy, radial and angular structure, and trajectory rollouts as complementary views of performance.Source: §V · p. 5
  • At r = 0.6, 1.0, and 1.5, dense/sparse errors were 8.8%, 0.8%, and 13.5%, versus 100.4%, 124.9%, and 747.2% for log-uniform.The dense/sparse sampler visits compact configurations more often while retaining mixed and extended geometries; its large-r accuracy is lower, though roughly an order of magnitude better than log-uniform sampling.Source: Figure 3 · p. 6, §V · p. 6

A. Isolated three-body interaction

  • For the pure-AT reference model with ν = 1.0 over r ∈[0.6, 5.0], the study compares learned and analytic dynamics.The evaluation uses potential landscapes and force fields, functional geometric dependences, and trajectory rollouts.Source: §V · p. 5, A. Isolated three-body interaction · p. 6

1. Potential landscapes and force fields

  • Across a two-dimensional configuration-space slice, relative L1 errors were 1.3% for potential and 1.1% for acceleration magnitude, with attractive and repulsive field regions reproduced.The near-field band at r ∈[0.6, 1.2] has relative potential residuals of approximately 0.9–1.0%; potential comparisons are centered because the zero of potential energy is arbitrary.Source: §1 · p. 6

2. Radial dependence

Scale and angle sweeps recover the Axilrod–Teller geometry closely, while pairwise controls and trajectory rollouts probe the model’s behavior beyond those field-level tests.

2. Radial dependence

  • The same representation fits pure Lennard–Jones with 0.45% potential and 0.76% acceleration relative L1 error, and Newtonian gravity with 2.6% mean relative acceleration error.For gravity, the distances-only ablation is indistinguishable at 2.5% mean relative acceleration error and R2 = 0.9997.Source: §2 · p. 7
  • 0.25% potential and 0.66% acceleration relative L1 errors across five reference geometries; the far-particle case separates near the numerical floor beyond the training ceiling.The far-particle relative errors are 87.7% for potential and 21.6% for acceleration, attributed to the vanishing three-body signal rather than a functional-form discrepancy.Source: §2 · pp. 6–7, Figure 5 · p. 8

3. Angular dependence

  • At fixed length scale, angular sweeps recover the potential and acceleration with mean relative errors of 0.81% and 1.06%, including the transition between repulsive and attractive branches.The swept angle changes triangle shape while holding the characteristic radial scale fixed.Source: §3 · p. 8, Figure 8 · p. 9

4. Trajectory rollouts

  • Across 100 rollouts, median center-of-mass-corrected position error is 1.9% of typical particle separation, rising to 7.1% by the window’s end.Center-of-mass drift reaches only 5.2 × 10^-6 by the end of the window.Source: §4 · p. 9
  • Median relative energy error is 11.5% over the rollout window and rises to 26.6% by its end, although the learned model conserves its own energy to integrator precision.The learned rollout’s energy is constant to within approximately 10^-6 relative error; the reported discrepancy compares against the true analytic energy.Source: §4 · pp. 9–11
  • The rollouts include unbound attractive trajectories that curve before separating, bound attractive trajectories that spiral into collision, and repulsive trajectories that straighten into free separation.The repulsive case’s curvature is confined to a brief initial window.Source: §4 · pp. 9–10

5. The canonical configurations of Axilrod and Teller

The learned potential reproduces the correct sign and qualitative dynamics in all four canonical triangle geometries used to illustrate the Axilrod–Teller interaction.

5. The canonical configurations of Axilrod and Teller

  • The learned potential gives the correct sign and qualitative dynamics for equilateral, right-angled, collinear, and far-removed-atom configurations.Their respective angular factors are 11/8, 1, −2, and, in the far-removed-atom limit, set by 1 − 3 cos^2 θ1.Source: §5 · p. 11

B. Three-body corrections on a pairwise background

On a pairwise Lennard–Jones background, the paper assesses whether the three-body contribution to acceleration exceeds the pairwise model’s fitting-error floor.

B. Three-body corrections on a pairwise background

  • Identifiability is framed as whether the omitted Axilrod–Teller acceleration exceeds the Lennard–Jones-only model’s residual error floor.The comparison uses the AT contribution, linear in its coupling, against the pairwise model’s residual on dynamics it was trained to reproduce.Source: B. Three-body corrections on a… · p. 11
  • For argon-reference configurations with rmin ≥ 0.9, the median pairwise-model error floor is approximately 0.045 reduced acceleration units.The comparison is motivated by asking when a three-body correction leaves a dynamical signature a learned model can resolve.Source: B. Three-body corrections on a… · p. 11

1. Physical coupling: a null result

On the Lennard–Jones background, the three-body term is resolved at amplified coupling but not at argon’s physical coupling with the selected model and data. The paper also tests sensitivity to the inner cutoff and uses symbolic regression to recover the interaction’s analytic structure.

1. Physical coupling: a null result

  • At argon’s physical coupling, the AT signal lies below the fitting floor for 85% of configurations, and the full LJ+AT model does not improve the median residual.The median residuals are ≈0.091 for LJ+AT and ≈0.053 for LJ-only; the null result is specific to the selected architecture and data budget.Source: §1 · p. 11

B. Three-body corrections on a pairwise background

  • The identifiability boundary is bracketed between ˜ν = 0.1 and 1.0: the three-body term is recovered when its signal clears the model’s error floor.At physical argon coupling its median signal is ≈0.009, below the pairwise floor; at ˜ν = 1.0 the full model’s median residual is 2.5× smaller than the pairwise model’s.Source: §2 · p. 11, §1 · p. 11

2. Amplified coupling

  • At ˜ν = 1.0, the AT contribution has median magnitude ≈0.087 and the LJ+AT model’s median residual (≈0.065) is 2.5× smaller than the LJ-only model’s (≈0.162).The amplified signal stands above the pairwise floor over most of configuration space.Source: §2 · p. 11

3. Sensitivity to the inner cutoff

  • Reducing the inner cutoff from rmin = 0.9 to 0.7 raises validation error from ≈0.07 to 2–3 and prevents clean convergence.Configurations on the steep LJ repulsive wall dominate fitting, leaving the physical AT signal about two orders of magnitude below the resulting error floor; the authors therefore use rmin = 0.9 by default for LJ systems.Source: §3 · p. 11

VI. SYMBOLIC RECOVERY OF THE INTERACTION LAW

  • Symbolic regression uses a stabilized relative-error objective to fit expressions to the learned potential over its geometric features.The prediction error is normalized by target magnitude plus an offset of ϵ = 0.3; standard squared error instead lets the small-r core dominate and returns a degenerate fit.Source: §VI · p. 12
  • As a sanity check, the same symbolic search recovers the analytic VAT exactly, with coefficients within 10−4 of their true values.This confirms that the feature set, operator set, and objective can recover the known law before they are applied to the learned potential.Source: §VI · p. 12
  • The recovered expression reproduces the AT angular numerator and r−9 radial law, with the correct relative weight and coupling, while matching the learned potential at R2 ≈ 1.0.A small final term breaks exact exchange symmetry but contributes at most ≲1.3% of |Vθ|; withholding angle cosines yields only R2 ≈ 0.4 against the analytic potential, whereas adding them recovers VAT exactly under the same operator budget.Source: §VI · pp. 12–13

VII. DISCUSSION AND CONCLUSIONS

The paper shows that physically structured LNNs can recover an irreducible three-body interaction from motion data. Data design and symbolic regression support recovery of the interaction’s structure, while the composite-system results delimit when the signal is identifiable.

VII. DISCUSSION AND CONCLUSIONS

  • The approach recovers an irreducible three-body interaction from motion data, with the composite LJ+AT results showing recovery only when the signal exceeds the learned model’s residual error.At argon’s more realistic coupling the signal remains below the error floor for the selected data and architecture.Source: §VII · p. 13
  • Sampling must cover relevant radial and angular regimes while allocating data to informative regions and avoiding near-singular and dynamically irrelevant far-field configurations.The authors report that these choices determine whether the steep radial dependence and angular structure can be learned.Source: §VII · p. 13
  • Symbolic regression on the learned representation recovers the AT interaction’s analytic structure and amplitude, including its ν ≃ 1 coupling, without an assumed functional form.The learned potential itself supplies the target for symbolic inference.Source: §VII · p. 13
  • The approach’s applicability beyond the studied setting depends on assumptions including a position-only potential and constant mass matrix.Within those boundaries, the authors conclude that trajectory data can support both reproduction of dynamics and recovery of underlying physics.Source: §VII · p. 13

Appendix A: Dataset generation

The appendix describes shape-space sampling that emphasizes strong-force configurations while retaining broader coverage, and explains how metrics address errors near zero-valued targets.

Appendix A: Dataset generation

  • Distances are sampled from a dense-region-weighted log-uniform mixture, while random rotations, translations, and label permutations enforce symmetry and preserve weak-interaction coverage.Velocities seed trajectory evaluation but enter training only through the fixed kinetic term and Euler–Lagrange algebra, not the potential network.Source: Appendix A · p. 13
  • Recorded position–acceleration pairs can be used after resampling, but practical cost depends on capturing close encounters and finite-difference noise can grow near them.Trajectory data under-represent compact, strong-force geometries because trajectories spend less time there.Source: Appendix A · p. 14

Appendix B: Training details

Appendix C: Evaluation metrics

  • MRE averages pointwise relative errors and can be dominated by near-zero targets, so the authors instead report magnitude-weighted relative L1 error where this instability occurs.The weighted metric down-weights small true values; Table IV presents both metrics for the two evaluations with the largest difference.Source: Appendix C · p. 14
  • Table IV compares MRE and relative L1 error per reference shape for radial dependence in the pure Axilrod–Teller system.For single fit-quality diagnostics, the authors use R2, where 1 is a perfect fit and 0 is no better than predicting the mean.Source: Table IV · p. 14, Appendix C · p. 14

Appendix D: Robustness of the symbolic recovery

The appendix reports robustness checks, including a comparison of radial decay rates, and discusses operator-set dependence and non-unique symbolic forms.

Appendix D: Robustness of the symbolic recovery

  • Appendix D supplies checks omitted from the main text, including numerical evidence distinguishing the r^-6 and r^-9 decay rates; Table V gives the pure-LJ control comparison.Source: Appendix D · pp. 14–15, Table V · p. 15

1. Operator-set dependence

  • The operator-set dependence check repeats symbolic search after removing the cube operator, testing whether the available operators predispose recovery toward the Axilrod–Teller form.The passage describes the test setup but does not state its outcome.Source: §1 · p. 15

2. Non-uniqueness.

  • Because triangle angle cosines obey an identity, distinct cosine-polynomial expressions can match every physical triangle, so search results need not be algebraically unique.The authors verify equivalence by reducing expressions to a common canonical form.Source: §2 · p. 15
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