Recovering Irreducible Three-Body Interactions with Lagrangian Neural Networks

The gist: Lagrangian Neural Networks can recover an irreducible three-body interaction from motion data alone by parameterizing a scalar Lagrangian and learning only the potential, which has been shown to reproduce the Axilrod-Teller interaction.

How it works

Lagrangian Neural Networks (LNNs) learn mechanical systems 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 system dynamics are obtained by differentiating this function through the Euler–Lagrange equations. The authors address whether LNNs can recover an irreducible many-body interaction from motion alone, focusing on the Axilrod-Teller triple-dipole interaction They prescribe the kinetic energy exactly and learn only the potential as a function of interparticle distances and angles.

Physical System and Interaction Model

The study considers three identical particles of mass m in two spatial dimensions with total potential V = X i<j VLJ(r ij) + VAT, where VLJ(r) is the pairwise Lennard-Jones potential. The interaction model is inspired by classical many-body fluid Hamiltonians incorporating both pairwise Lennard-Jones and three-body Axilrod–Teller contributions. The interaction has analytic virtues, as its radial and angular structure are known in closed form, and its geometric content, which is an angular factor that changes sign with the shape of the particle triangle, is a good study case for the kind of many-body structure that a pairwise model is provably unable to represent.

Training Methodology and Data Design

The training methodology involves minimizing the mean absolute error of predicted accelerations, Ltrain = 1/N X N i=1 q¨ pred i - q¨ true i. The sampling strategy is designed to concentrate training data where the three-body signal is strongest while preserving broad coverage of configuration space. Specifically, the dense/sparse sampler maximizes exposure to the narrow regime where the interaction remains both informative and numerically stable. The input representation includes augmented coordinates x = (r12, r23, r31, cos θ1, cos θ2, cos θ3), which exposes radial and angular structure explicitly.

Evaluation and Symbolic Recovery

The learned potential is evaluated at three levels: field-level accuracy, functional geometric structure, and trajectory rollouts for the isolated AT system. The symbolic regression on the trained potential returns the Axilrod-Teller expression with angular numerator, radial law, and coupling constant intact. The study investigates identifiability by comparing the AT contribution against the model’s own fitting-error floor in a Lennard-Jones background.

Key Findings on Identifiability and Structure

The research demonstrates that 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. The learned potential reproduces the radial scaling, angular dependence, and characteristic sign reversal of the Axilrod-Teller interaction. Furthermore, symbolic regression recovers a candidate functional form of Vθ that matches the analytic VAT with R squared ≈ 1.0. The angular coordinates are shown to provide essential representational structure, as withholding them causes the search to fail in reconstructing Φ from distances alone.

Conclusion

The study establishes a framework where physically structured deep learning can serve as a tool for scientific inference by recovering an explicit irreducible three-body potential from trajectories data. This combination of LNN-based dynamical recovery and symbolic inference is among the first demonstrations of recovering an explicit irreducible three-body potential in this way. The methodology provides a route toward studying other systems in which higher-order interactions are difficult to formulate or their structure is not known a priori. The applicability of the results depends on the validity of assumptions such as a position-only potential and constant mass matrix within these boundaries. The trajectory data can contain enough information to both reproduce dynamics and recover the physics underlying them.

1. textbfIncrease Model Interpretability via Physics-Informed Structure Preservation: The LNN framework incorporates the principle of stationary action into the model structure. This allows for a domain-specific variant of a general Lagrangian neural network, in which a priori physical knowledge is incorporated analytically, making the model substantially more interpretable than a fully black-box dynamical model while preserving the flexibility of deep learning.

2. textbfEnable Recovery of Unknown Interaction Laws: The system can recover an irreducible three-body interaction from trajectories alone by using symbolic regression on the trained potential, with no assumed functional form, leading to a closed-form law that reveals what interaction the model has actually learned.

3. textbf Improve Data Efficiency via Adaptive Sampling: Implement a signal-aware sampling strategy that concentrates training data where the three-body signal is strongest while preserving broad coverage of configuration space. This ensures the sampler focuses on the narrow regime where the interaction remains both informative and numerically stable, leading to significantly lower relative acceleration errors compared to plain log-uniform sampling.

4. textbf Achieve Robust Identifiability Testing: The system can distinguish between a null result (where the three-body term is below the model’s fitting error floor) and an amplified coupling scenario where it is resolvable, by comparing the AT contribution against the model’s own fitting-error floor.

5. textbf Provide Geometry-Aware Feature Representation: The input representation utilizes pairwise distances augmented with angle cosines, which exposes both radial and angular structure explicitly, acting as an inductive bias that has been used in machine-learned interatomic potentials to ensure the network captures the necessary geometric information for three-body potentials.

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.