[Paper Review] Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe

Hong Qin|arXiv (Cornell University)|Oct 22, 2019

Model Reduction and Neural Networks83 references30 citations

TL;DR

This paper proposes a machine learning framework that learns discrete field theories directly from observational data on a spacetime lattice, bypassing the need to know underlying physical laws. The method trains a discrete Lagrangian density using neural networks and uses it to predict new physical behaviors—such as elliptical, parabolic, and hyperbolic planetary orbits—without prior knowledge of Newton’s laws, demonstrating strong generalization and structure-preserving accuracy.

ABSTRACT

A method for machine learning and serving of discrete field theories in physics is developed. The learning algorithm trains a discrete field theory from a set of observational data on a spacetime lattice, and the serving algorithm uses the learned discrete field theory to predict new observations of the field for new boundary and initial conditions. The approach to learn discrete field theories overcomes the difficulties associated with learning continuous theories by artificial intelligence. The serving algorithm of discrete field theories belongs to the family of structure-preserving geometric algorithms, which have been proven to be superior to the conventional algorithms based on discretization of differential equations. The effectiveness of the method and algorithms developed is demonstrated using the examples of nonlinear oscillations and the Kepler problem. In particular, the learning algorithm learns a discrete field theory from a set of data of planetary orbits similar to what Kepler inherited from Tycho Brahe in 1601, and the serving algorithm correctly predicts other planetary orbits, including parabolic and hyperbolic escaping orbits, of the solar system without learning or knowing Newton's laws of motion and universal gravitation. The proposed algorithms are also applicable when effects of special relativity and general relativity are important. The illustrated advantages of discrete field theories relative to continuous theories in terms of machine learning compatibility are consistent with Bostrom's simulation hypothesis.

Motivation & Objective

• To develop a machine learning method that discovers discrete field theories from observational data without prior knowledge of physical laws.
• To overcome the challenges of learning continuous field theories via differential equations by shifting to discrete field theories.
• To create a serving algorithm that preserves geometric structures (e.g., symplecticity, energy conservation) for long-term accuracy.
• To demonstrate the method’s ability to predict complex physical behaviors—such as escaping orbits—using only observational data.
• To explore the compatibility of discrete field theories with the simulation hypothesis, suggesting nature may operate via discrete, computable laws.

Proposed method

• Formulates field theories in terms of discrete Lagrangian densities $ L_d $, which depend on the field values at $ n+1 $ adjacent spacetime points.
• Trains the discrete Lagrangian using a neural network on observational data, avoiding the need to compute second-order derivatives.
• Employs structure-preserving geometric algorithms for serving the learned theory, ensuring long-term fidelity and conservation laws.
• Uses a variational principle on the discrete action to derive evolution rules that preserve symplectic and momentum structures.
• Applies the learning algorithm to data from planetary orbits (Mercury to Jupiter) to infer a discrete field theory of gravitational dynamics.
• Validates predictions on unseen boundary conditions, including parabolic and hyperbolic trajectories, without prior knowledge of Newtonian mechanics.

Experimental results

Research questions

• RQ1 Can a machine learning algorithm discover a discrete field theory from observational data alone, without prior knowledge of physical laws?
• RQ2 How does learning discrete field theories compare to learning continuous differential equations in terms of data efficiency and generalization?
• RQ3 Can a learned discrete field theory predict non-trivial physical behaviors—such as escaping orbits—beyond the training data?
• RQ4 To what extent do structure-preserving geometric algorithms enhance the predictive accuracy of learned field theories?
• RQ5 Does the success of discrete field theory learning support the simulation hypothesis, where the universe operates as a discrete computational system?

Key findings

• The learning algorithm successfully reconstructed a discrete field theory from observational data of planetary orbits similar to Kepler’s data from Tycho Brahe.
• The serving algorithm accurately predicted parabolic and hyperbolic orbits—behaviors not present in the training data—without any knowledge of Newton’s laws.
• The method achieved long-term stability and energy conservation due to the use of structure-preserving geometric algorithms in the serving phase.
• The discrete Lagrangian approach avoided the need to compute second-order derivatives, simplifying training and improving robustness.
• The results suggest that discrete field theories are more amenable to machine learning than continuous ones, aligning with Bostrom’s simulation hypothesis.
• The framework is extendable to relativistic systems, as demonstrated by its applicability to problems involving special and general relativity.