Advancing Solutions for the Three-Body Problem Through Physics-Informed Neural Networks

TL;DR

The paper tackles the chaotic Three-Body Problem and proposes physics-informed neural networks (PINNs) as regularized solvers that embed Newtonian dynamics into learning. By combining data-driven losses with a PDE residual term, the authors compare non-autoregressive and autoregressive DNN/ResNet architectures, guided by an alpha-weighted physics loss $\mathcal{L}=\mathcal{L}_{MAE}^{(u)}+\alpha\mathcal{L}^{(f)}$. They show that physics-informed models reduce coherence errors and that ResNet-based PINNs offer favorable balance between physical consistency and predictive accuracy, albeit with longer training times. The results indicate PINNs can yield accurate, time-efficient TBP solutions and demonstrate the value of priors for chaotic dynamical systems, with recommendations for training dynamics and future physics-aware architectures.

Abstract

First formulated by Sir Isaac Newton in his work "Philosophiae Naturalis Principia Mathematica", the concept of the Three-Body Problem was put forth as a study of the motion of the three celestial bodies within the Earth-Sun-Moon system. In a generalized definition, it seeks to predict the motion for an isolated system composed of three point masses freely interacting under Newton's law of universal attraction. This proves to be analogous to a multitude of interactions between celestial bodies, and thus, the problem finds applicability within the studies of celestial mechanics. Despite numerous attempts by renowned physicists to solve it throughout the last three centuries, no general closed-form solutions have been reached due to its inherently chaotic nature for most initial conditions. Current state-of-the-art solutions are based on two approaches, either numerical high-precision integration or machine learning-based. Notwithstanding the breakthroughs of neural networks, these present a significant limitation, which is their ignorance of any prior knowledge of the chaotic systems presented. Thus, in this work, we propose a novel method that utilizes Physics-Informed Neural Networks (PINNs). These deep neural networks are able to incorporate any prior system knowledge expressible as an Ordinary Differential Equation (ODE) into their learning processes as a regularizing agent. Our findings showcase that PINNs surpass current state-of-the-art machine learning methods with comparable prediction quality. Despite a better prediction quality, the usability of numerical integrators suffers due to their prohibitively high computational cost. These findings confirm that PINNs are both effective and time-efficient open-form solvers of the Three-Body Problem that capitalize on the extensive knowledge we hold of classical mechanics.

Figures (14)

• Figure 1: Simplified schemata of a , illustrating the difference to classical . The section to the left of the red vertical line depicts a standard .
• Figure 2: Illustration of the particle's reference plane, where the green area represents all the valid positions where $\mathbf{p}_2$ can be initially placed. The red point represents the singularity, a point where $\mathbf{p}_2$'s and $\mathbf{p}_3$'s positions coincide. Figure adapted from Breen2020.
• Figure 3: Visualization of the dependence of the BRUTUS three-body simulations on the choice of initial position for particle $2$.
• Figure 4: Residual block, where $\sigma$ represents any nonlinear activation function and a Fully Connected Linear Layer is a layer of neurons where each is connected to each of the inputs and outputs. A ResNet is composed of successive applications of this block.
• Figure 5: Error comparison over time between autoregressive and non-autoregressive models, highlighting the exponential error growth of autoregressive models.