Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe

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.

Figures (12)

• Figure 1: Spacetime lattice and discrete field $\psi$. The discrete Lagrangian density $L_{d}(\psi_{i,j},\psi_{i+1,j},\psi_{i,j+1})$ of the grid cell whose lower left vertex is at the grid point $(i,j)$ is chosen to be a function of the values of the discrete field at the three vertices marked by solid circles. The action $\mathscr{\mathcal{A}}_{d}$ of the system depends on $\psi_{i,j}$ through the discrete Lagrangian densities of the three neighboring grid cells indicated by gray shading.
• Figure 2: The predicted sequence $\psi_{i}$ (solid circles) from the learned discrete field theory and the training sequence $\bar{\psi}_{i}$ (empty squares) are barely distinguishable in the figure. The discrete Lagrangian is trained until the loss function $F(\overline{\psi})$ is less than $10^{-7}$.
• Figure 3: The predicted time sequence (solid circles) agrees with the time sequence (empty squares) accurately solved for from the nonlinear ODE (\ref{['ex1ODE']}). The dynamics starts at $\psi_{0}=-0.6$, and its characteristics is significantly different from that of the time sequence in Fig. \ref{['fig:ex1traning']}.
• Figure 4: Starting at a much smaller amplitude, i.e., $\psi_{0}=0.1$, the predicted sequence (solid circles) shows the behavior of linear oscillation, in contrast with the strong nonlinearity of the sequence in Fig. \ref{['fig:ex1traning']} and the mild nonlinearity of the sequence in Fig. \ref{['fig:ex1test1']}. The predicted time sequence agrees with the time sequence (empty squares) accurately solved for from the nonlinear ODE (\ref{['ex1ODE']}).
• Figure 5: The training sequence (empty squares) represents a nonlinear oscillation in potential wall between $\psi=\pm1.6$ in Fig. \ref{['fig:ex2V']}. The trained discrete Lagrangian density $L_{d}$ is accepted when the loss function $F(\overline{\psi})$ on the training sequence is less than $10^{-7}$. The predicted sequence (solid circles) from the learned discrete field theory agrees very well with the training sequence.

Theorems & Definitions (2)

• Example 1
• Example 2