Comparing Physics-Informed and Physics-Enforced Neural Networks for the Chaotic Three-Body Problem

Abstract

The three-body gravitational problem is a canonical benchmark for chaotic dynamical systems, admitting no general closed-form solution and exhibiting sensitive dependence on initial conditions. We present a controlled comparative benchmark of four neural network architectures - a Deep Neural Network (DNN), a Physics-Informed Neural Network (PINN), a Hamiltonian Neural Network (HNN), and a Lagrangian Neural Network (LNN) - for trajectory prediction in the planar equal-mass three-body problem. All models are trained on 1,000 high-precision trajectories generated using the DOP853 integrator and evaluated on trajectory accuracy, long-horizon generalisation, and energy conservation. To the best of our knowledge, this is the first systematic evaluation of HNNs and LNNs as trajectory predictors for the gravitational three-body problem. The DNN achieves the lowest in-horizon error (mean absolute error: 0.3166), while the LNN surpasses all models in long-horizon generalisation and energy conservation. The PINN achieves the best ordinary differential equation compliance but the worst trajectory accuracy, demonstrating the challenge of loss balancing in chaotic systems. These results, obtained under deliberately controlled but limited conditions (1,000 trajectories, equal-mass planar case), provide initial evidence that Lagrangian structural bias offers a measurable advantage for this benchmark, motivating larger-scale evaluation as a next step.