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Recovering Irreducible Three-Body Interactions with Lagrangian Neural Networks

This paper demonstrates that Lagrangian Neural Networks, when combined with signal-aware sampling and symbolic regression, can successfully recover the closed-form expression of an irreducible three-body Axilrod-Teller interaction directly from trajectory data, even in the presence of pairwise forces.

Original authors: Sena Kalabalik, Sukru Caglar, Abdullah Umut Hamzaogullari, Arkadas Ozakin

Published 2026-10-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sena Kalabalik, Sukru Caglar, Abdullah Umut Hamzaogullari, Arkadas Ozakin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, scientists have long relied on a powerful idea to describe how things move: the principle that nature always chooses the path of least resistance. This concept, known as the principle of stationary action, allows researchers to predict the future motion of objects by understanding a single mathematical quantity called the Lagrangian. For centuries, this framework has successfully explained everything from swinging pendulums to orbiting planets, but it has traditionally assumed that interactions between objects are simple and additive. In this view, the force between three objects is just the sum of the forces between each pair. However, the real world is often more complex. Sometimes, three objects interact in a way that cannot be broken down into pairs; the presence of a third object fundamentally changes how the other two behave, creating a unique, irreducible relationship that depends on the specific shape they form together.

A team of researchers at Boğaziçi University in Turkey set out to see if modern artificial intelligence could discover these complex, three-way interactions on its own, simply by watching how particles move. They turned to a specific type of machine learning model called a Lagrangian Neural Network. Unlike standard computer programs that try to memorize patterns, these networks are built with the laws of physics baked into their very structure. They are designed to learn the underlying rules of motion by observing trajectories—the paths particles take over time—without being told what the forces are. The researchers wanted to know if such a system could look at the motion of three particles and figure out that they were being pushed or pulled by a force that only exists when all three are present, a force that changes its nature depending on whether the particles form a triangle, a line, or some other shape.

To test this, the scientists created a simulated environment containing three identical particles. They programmed the computer to generate millions of different movements for these particles based on a known, complex force called the Axilrod-Teller interaction. This force is a real physical phenomenon that occurs between neutral atoms, where the energy depends not just on how far apart the atoms are, but on the angles they form with one another. If the atoms form a compact triangle, the force pushes them apart; if they stretch into a long, thin line, the force pulls them together. The researchers then fed the computer only the recorded paths of these moving particles, hiding the formula for the force entirely. The neural network had to work backward from the motion to deduce the invisible rules governing it.

The results were striking. The artificial intelligence successfully learned the hidden force. It did not just memorize the paths; it reconstructed the actual mathematical law that generated them. The network discovered that the force followed a specific pattern: it grew incredibly strong as the particles got closer, and it flipped from repulsive to attractive based on the shape of the triangle they formed. To ensure the system wasn't just guessing, the researchers checked its predictions against the known physics. The network's learned force matched the real force with high precision, correctly reproducing the way the interaction changes with distance and angle. Even more impressively, when the researchers asked the computer to translate its internal, complex mathematical understanding back into a simple, human-readable formula, it did so with remarkable accuracy. The computer wrote down the exact expression for the three-body interaction, including the specific terms that describe the angular dependence and the correct coupling strength. While the recovered formula was not mathematically perfect—showing a very small residual error and a minor term that slightly broke the perfect symmetry of the particles—it captured the core physics with such fidelity that it effectively revealed the true rulebook.

However, the study also revealed a crucial limit to this kind of discovery. In the real world, this three-body force is usually a tiny correction to the much stronger forces that act between pairs of atoms. The researchers tested whether their system could find this tiny signal when it was buried under the noise of the larger, simpler forces. They found that if the three-body force was too weak compared to the errors inherent in the computer's learning process, the system could not detect it. The signal was simply lost in the background. But when they amplified the strength of the three-body interaction, making it large enough to stand out clearly above the noise, the system found it immediately. This suggests that while artificial intelligence can uncover complex physical laws from motion, it requires the signal to be strong enough to be distinguished from the limitations of the model itself.

This work demonstrates a new path for scientific discovery. By combining deep learning with the fundamental structure of physics, researchers can now move beyond simply predicting how a system will behave to actually discovering the laws that govern it. The study shows that if you provide the right kind of data and the right kind of mathematical framework, a machine can look at the chaotic dance of particles and write down the rulebook for their interaction, even when that rulebook describes a relationship that involves three objects acting as a single, inseparable unit.

Problem Statement
Lagrangian Neural Networks (LNNs) have demonstrated success in learning mechanical systems from trajectory data by parameterizing a scalar Lagrangian and deriving dynamics via the Euler-Lagrange equations. However, prior work has largely been restricted to pairwise-additive interactions. A critical open question remains whether LNNs can recover irreducible many-body interactions—potentials that cannot be decomposed into a sum of pairwise terms—from motion data alone. This paper addresses this gap by investigating the recovery of the Axilrod-Teller (AT) triple-dipole interaction, the leading three-body dispersion term. The AT potential depends on the collective geometry of three particles (specifically the angles of the triangle they form) in a way that cannot be represented by any sum of pairwise functions, presenting a qualitative challenge to standard pairwise-decomposition architectures.

Methodology
The authors propose a framework that combines domain-specific physical knowledge with deep learning to isolate and learn the three-body potential.

Fixed-Kinetic Architecture: Instead of learning the entire Lagrangian $L(q, \dot{q})$ , the authors prescribe the kinetic energy $T$ analytically (assuming known particle masses and Cartesian coordinates) and use a neural network $V_\theta$ to learn only the potential energy. This reduces the learning task to recovering a scalar function from its gradient field, eliminating the burden of inferring the known quadratic kinetic structure.
Augmented Input Representation: The potential network receives an input vector $x = (r_{12}, r_{23}, r_{31}, \cos \theta_1, \cos \theta_2, \cos \theta_3)$ , comprising pairwise distances and the cosines of the interior angles. While the angles are mathematically determined by the distances, this "geometry-augmented" representation acts as an inductive bias, explicitly exposing the radial and angular structure required to capture the AT interaction.
Signal-Aware Sampling: Recognizing that the AT force scales as $r^{-10}$ , the authors employ a specialized sampling strategy. They use a dense/sparse mixture to concentrate training data in the "strong-interaction" regime (small separations) where the three-body signal is most informative, while maintaining broad coverage of the configuration space. This prevents the loss function from being dominated by noise in the far-field or failing to fit the steep near-field.
Training Objective: The network is trained by minimizing the mean absolute error between predicted accelerations (derived via automatic differentiation of the learned Lagrangian) and reference accelerations. The potential is never supervised directly; it is inferred implicitly through acceleration matching.
Symbolic Regression: To test if the learned neural representation can be converted into an explicit physical law, the authors apply symbolic regression (using the PySR library) to the trained potential $V_\theta$ , searching for a closed-form expression without assuming the functional form a priori.

Key Results
The study evaluates the approach across three levels: field-level accuracy, functional geometric structure, and trajectory stability.

Isolated AT System:
- Field-Level Accuracy: The learned potential reproduces the analytic potential landscape and acceleration field with high fidelity (relative $L_1$ error $\approx 1.3\%$ for potential and $1.1\%$ for acceleration magnitude).
- Geometric Dependence: The model correctly recovers the $r^{-9}$ radial scaling and the characteristic angular dependence, including the sign reversal of the interaction (repulsive for equilateral triangles, attractive for collinear configurations).
- Controls: When trained on purely pairwise systems (Lennard-Jones or Newtonian gravity) using the same angular-augmented inputs, the network does not learn spurious angular dependencies, confirming the representation is neutral for purely radial interactions.
- Trajectory Rollouts: The learned dynamics generate stable trajectories. While long-term energy drift occurs due to the compounding of small local errors in steep potential regions, the model preserves momentum and center-of-mass conservation to high precision.
Composite LJ+AT System (Identifiability):
- The authors investigate whether the three-body term is identifiable when embedded in a dominant pairwise Lennard-Jones background.
- Physical Coupling ( $\tilde{\nu} \approx 0.1$ ): At the coupling strength typical for argon, the three-body signal is roughly five times smaller than the irreducible fitting error (noise floor) of the pairwise model. Consequently, the three-body term is not identifiable; adding the AT term does not improve the model's fit to the dynamics.
- Amplified Coupling ( $\tilde{\nu} = 1.0$ ): When the coupling is amplified, the AT signal exceeds the model's error floor. In this regime, the full model successfully resolves the three-body contribution, reducing the residual error significantly compared to a pairwise-only model.
Symbolic Recovery:
- Applying symbolic regression to the trained potential $V_\theta$ successfully recovers the closed-form Axilrod-Teller expression: $V_{SR} \propto \frac{1 + 3 \cos \theta_1 \cos \theta_2 \cos \theta_3}{(r_{12}r_{23}r_{31})^3}$ .
- The regression correctly identifies the angular numerator, the $r^{-9}$ radial law, and the coupling constant. Crucially, when the angular inputs ( $\cos \theta_i$ ) are withheld during symbolic regression, the algorithm fails to reconstruct the correct structure, providing independent evidence that the angular features are essential for representing the interaction.

Significance and Claims
The paper claims to demonstrate that Lagrangian Neural Networks, when combined with physically informed sampling and domain-specific structural constraints, can recover an irreducible three-body interaction from trajectory data alone.

Scientific Inference: The work establishes a framework where deep learning serves as a tool for scientific inference. By encoding known physics (kinetic terms) and using targeted data design, the model can isolate complex interactions.
From Data to Law: The successful application of symbolic regression to recover the exact analytic form of the AT potential (including its geometric dependence and coupling) from a neural network trained solely on trajectory data represents a step toward extracting interpretable physical laws from dynamical data.
Limits of Identifiability: The study provides a quantitative measure of the sensitivity of trajectory-based recovery, showing that higher-order interactions are only recoverable when their dynamical signature exceeds the residual modeling error of the underlying architecture.
Generalizability: While the specific results concern the AT interaction, the methodology (fixed kinetic terms, augmented inputs, signal-aware sampling) is presented as a transferable approach for studying other systems where higher-order interactions are difficult to formulate or their structure is unknown a priori.

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