Recovering Irreducible Three-Body Interactions with Lagrangian Neural Networks
Authors: Sena Kalabalik, Sukru Caglar, Abdullah Umut Hamzaogullari, Arkadas Ozakin
Organizations: Department of Physics, Bo˘gazi¸ci University, Istanbul, Türkiye · Department of Computer Engineering, Bo˘gazi¸ci University, Istanbul, Türkiye
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.
Figures & tables
Figure 1: The AT interaction at a glance: analytic forces on all three particles for representative triangle shapes (rows) at increasing size (columns), with arrow color encoding force magnitude. Compact, near-equilateral shapes ( Φ>0 ) are pushed apart; elongated, near-collinear shapes ( Φ<0 ) are pulled together; the overall strength falls dramatically as r−10 in force.
Figure 2: Schematic of the fixed-kinetic Lagrangian network: known structure (kinetic term, symmetries) is encoded exactly, and only the interaction potential is learned. The potential is not supervised directly, the training signal reaches it only through the accelerations it generates.
Method
Learned components
DeLaN [ 2 ]
Mθ(q) and Vθ(q)
LNN [ 1 ]
entire Lθ(q,q˙)
Hamzaoğulları & Ozakın [ 4 ]
entire Lθ(q,q˙) ,
Hessian-regularized
This work
analytic T(q˙)+Vθ(x(q))
Table 1: Lagrangian parameterizations relevant to this work.
Figure 3: Relative acceleration error along an equilateral configuration sweep for models trained with the proposed dense/sparse sampler and with plain log-uniform sampling. The dense/sparse model reaches relative errors of 8.8% , 0.8% , and 13.5% at r=0.6 , 1.0 , and 1.5 , against 100.4% , 124.9% , and 747.2% for the log-uniform model over the same points. Both models degrade further into the tail, but the gap persists, an order-of-magnitude advantage holds even where both models are far from quantitatively accurate.
Figure 4: Field-level comparison for the isolated AT interaction. Two particles (white markers) are held fixed while the third is evaluated over a two-dimensional grid of positions. Top row: analytic potential, learned potential, and their absolute difference. The potentials are centered by subtracting their spatial mean, reflecting the arbitrary additive constant in the potential energy. Bottom row: corresponding acceleration fields shown as streamlines colored by magnitude, together with the pointwise acceleration residual. The region where any pairwise distance falls below the training cutoff rmin is masked. The pink overlay indicates regions that were sampled less densely during training. The two fixed particles are separated by r12=1.20 ; axes are in reduced units.
Figure 5: Radial dependence of the learned Axilrod–Teller interaction. Each column corresponds to a fixed reference triangle, with all remaining configurations obtained by uniformly scaling that geometry. The horizontal axis r is the overall scale factor multiplying a fixed unit shape, so every pairwise distance grows ∝r . For the equilateral family r is the side length, for the others the linear size of the reference triangle. Because rescaling leaves all angles fixed, each family must follow V∝r−9 , ∣a∣∝r−10 . The far-particle family is the sole visibly separating curve. As the third atom recedes –hence surpassing the ceiling distance in training(App. A )–, the net three-body force vanishes fastest, hence both curves ride the numerical/representational floor and their apparent divergence is the asymptotic-decoupling limit, an expected consequence of evaluating outside the training domain. (a) Learned and analytic potentials. (b) Corresponding acceleration magnitudes.
Figure 6: Pure Lennard–Jones control. The same geometric input representation, including both pairwise distances and angle cosines, is trained on a purely pairwise interaction. The learned potential remains in good agreement with the analytic LJ radial dependence across all reference geometries, indicating that the additional angular inputs do not introduce spurious shape dependence.
Figure 7: Newtonian gravity control. Learned versus analytic potential (after gauge-fixing by a single additive offset) over a set of random three-body configurations within the training range, for the full (r,cosθ) model and the distances-only ablation. Both recover the pairwise 1/r potential essentially exactly ( R2=0.9999 and 0.9997 ), with mean relative acceleration errors of 2.6% and 2.5% , confirming that the geometric featurization fits a radial law nothing like the Axilrod–Teller term’s, and that the angular inputs stay inert when the interaction carries no angular signal.
Figure 8: Angular dependence of the learned interaction at fixed length scale. Particle 1 is at the origin, particle 2 at (s,0) , and particle 3 at s(cosθ,sinθ) , so the two sides meeting at particle 1 are held equal, r12=r13=s , while the swept apex angle θ sets the third side r23=2ssin(θ/2) and hence the shape. Thus s fixes the characteristic radial scale while θ alone varies the triangle shape, isolating the angular factor Φ at (near-)constant size. (a) Potential, after subtracting its mean. (b) Corresponding acceleration magnitude.
Figure 9: Trajectory-level evaluation of the learned dynamics.
Figure 10: Reference (dashed) and learned (solid) dynamics initialized from the four triangle geometries used by Axilrod and Teller [ 5 ] to illustrate the sign of Φ : equilateral, right-angled, collinear, and the far-removed-atom limit.
Figure 11: Identifiability of the three-body term on a pairwise background, versus the inner separation rmin (medians per bin). (a) The analytic AT contribution to the acceleration, ∣aAT∣=ν~∣aAT(1)∣ (what the pairwise model omits) at the physical ( ν~=0.1 ) and amplified ( ν~=1.0 ) couplings, against the pairwise (LJ-only) model’s error floor. (b) Residual of the pairwise-only and full LJ+AT models against the true composite acceleration, at both couplings.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
System
rmin
rdense
rmax
pdense
AT
0.6
1.5
5.0
0.9
LJ
0.9
1.5
3.0
0.5
LJ + AT
0.9
1.5
3.0
0.5
Appendix
Table 2: Dataset-generation parameters.
Hyperparameter
Value
Optimizer
RAdam
Learning rate
10−3
Weight decay
10−6
Learning-rate scheduler
Cosine annealing
Scheduler period ( Tmax )
20 epochs
Minimum learning rate ( ηmin )
5×10−4
Appendix
Table 3: Training hyperparameters used throughout this work.
Shape
MRE ( V )
L1 ( V )
MRE ( a )
L1 ( a )
Equilateral
0.29%
0.32%
3.28%
1.78%
Right
0.42%
0.18%
5.73%
0.27%
Line
0.33%
0.22%
10.77%
0.88%
Far particle
194.59%
87.65%
8764.27%
21.59%
Isosceles
0.56%
0.22%
11.20%
0.20%
Scalene
0.74%
0.33%
11.62%
0.17%
Appendix
Table 4: Radial dependence (pure AT, r∈[rmin,1.6] ): MRE vs. relative L1 error, per reference shape. The far-particle family’s net force vanishes fastest as the third atom recedes, so a small absolute residual there becomes a large relative one under either metric, reported per-shape.
Shape
MRE ( V )
L1 ( V )
MRE ( a )
L1 ( a )
Equilateral
0.55%
0.53%
1.54%
1.23%
Right
1.17%
0.16%
0.34%
0.48%
Line
0.99%
0.52%
0.84%
0.99%
Far particle
1.56%
0.77%
7.39%
0.96%
Isosceles
1.51%
0.46%
0.46%
0.59%
Scalene
0.66%
0.27%
0.25%
0.33%
Appendix
Table 5: Same comparison for the pure-LJ control ( r∈[0.9,1.6] ). The far-particle divergence between MRE and L1 is far milder here than for AT (Table 4 ). Direct numerical evidence of the r−6 vs. r−9 decay-rate difference between the two interactions.
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