AI Feynman: a Physics-Inspired Method for Symbolic Regression

A core challenge for both physics and artificial intellicence (AI) is symbolic regression: finding a symbolic expression that matches data from an unknown function. Although this problem is likely to be NP-hard in principle, functions of practical interest often exhibit symmetries, separability, compositionality and other simplifying properties. In this spirit, we develop a recursive multidimensional symbolic regression algorithm that combines neural network fitting with a suite of physics-inspired techniques. We apply it to 100 equations from the Feynman Lectures on Physics, and it discovers all of them, while previous publicly available software cracks only 71; for a more difficult test set, we improve the state of the art success rate from 15% to 90%.Comment: 15 pages, 2 figs. Our code is available at https://github.com/SJ001/AI-Feynman and our Feynman Symbolic Regression Database for benchmarking can be downloaded at https://space.mit.edu/home/tegmark/aifeynman.htm

Dynamical inference from a kinematic snapshot: The force law in the Solar System

If a dynamical system is long-lived and non-resonant (that is, if there is a set of tracers that have evolved independently through many orbital times), and if the system is observed at any non-special time, it is possible to infer the dynamical properties of the system (such as the gravitational force or acceleration law) from a snapshot of the positions and velocities of the tracer population at a single moment in time. In this paper we describe a general inference technique that solves this problem while allowing (1) the unknown distribution function of the tracer population to be simultaneously inferred and marginalized over, and (2) prior information about the gravitational field and distribution function to be taken into account. As an example, we consider the simplest problem of this kind: We infer the force law in the Solar System using only an instantaneous kinematic snapshot (valid at 2009 April 1.0) for the eight major planets. We consider purely radial acceleration laws of the form a_r = -A [r/r_0]^{-\alpha}, where r is the distance from the Sun. Using a probabilistic inference technique, we infer 1.989 < \alpha < 2.052 (95 percent interval), largely independent of any assumptions about the distribution of energies and eccentricities in the system beyond the assumption that the system is phase-mixed. Generalizations of the methods used here will permit, among other things, inference of Milky Way dynamics from Gaia-like observations

Large-scale structure perturbation theory without losing stream crossing

We suggest an approach to perturbative calculations of large-scale clustering in the Universe that includes from the start the stream crossing (multiple velocities for mass elements at a single position) that is lost in traditional calculations. Starting from a functional integral over displacement, the perturbative series expansion is in deviations from (truncated) Zel'dovich evolution, with terms that can be computed exactly even for stream-crossed displacements. We evaluate the one-loop formulas for displacement and density power spectra numerically in 1D, finding dramatic improvement in agreement with N-body simulations compared to the Zel'dovich power spectrum (which is exact in 1D up to stream crossing). Beyond 1D, our approach could represent an improvement over previous expansions even aside from the inclusion of stream crossing, but we have not investigated this numerically. In the process we show how to achieve effective-theory-like regulation of small-scale fluctuations without free parameters.Comment: added pedagogical explanation of key math trick in appendi

Tidal Tails Test the Equivalence Principle in the Dark Sector

Satellite galaxies currently undergoing tidal disruption offer a unique opportunity to constrain an effective violation of the equivalence principle in the dark sector. Theories in which cold dark matter (CDM) couples to a light scalar field naturally lead to a long-range force between dark matter particles. An inverse-square-law force of this kind would manifest itself as a violation of the equivalence principle in the dynamics of CDM compared to baryons in the form of gas or stars. In a previous paper, we showed that an attractive force would displace stars outwards from the bottom of the satellite's gravitational potential well, leading to a higher fraction of stars being disrupted from the tidal bulge further from the Galactic center. Since stars disrupted from the far (near) side of the satellite go on to form the trailing (leading) tidal stream, an attractive dark-matter force will produce a relative enhancement of the trailing stream compared to the leading stream. This distinctive signature of a dark-matter force might be detected through detailed observations of the tidal tails of a disrupting satellite, such as those recently performed by the Two-Micron All-Sky Survey (2MASS) and Sloan Digital Sky Survey (SDSS) on the Sagittarius (Sgr) dwarf galaxy. Here we show that this signature is robust to changes in our models for both the satellite and Milky Way, suggesting that we might hope to search for a dark-matter force in the tidal features of other recently discovered satellite galaxies in addition to the Sgr dwarf.Comment: 29 pages, 13 figures, final version published in PR

The general relativistic two body problem

The two-body problem in General Relativity has been the subject of many analytical investigations. After reviewing some of the methods used to tackle this problem (and, more generally, the N-body problem), we focus on a new, recently introduced approach to the motion and radiation of (comparable mass) binary systems: the Effective One Body (EOB) formalism. We review the basic elements of this formalism, and discuss some of its recent developments. Several recent comparisons between EOB predictions and Numerical Relativity (NR) simulations have shown the aptitude of the EOB formalism to provide accurate descriptions of the dynamics and radiation of various binary systems (comprising black holes or neutron stars) in regimes that are inaccessible to other analytical approaches (such as the last orbits and the merger of comparable mass black holes). In synergy with NR simulations, post-Newtonian (PN) theory and Gravitational Self-Force (GSF) computations, the EOB formalism is likely to provide an efficient way of computing the very many accurate template waveforms that are needed for Gravitational Wave (GW) data analysis purposes.Comment: 43 pages, 4 figures, to appear in the Brumberg Festschrift, edited by S. M. Kopeikein, and to be published by de Gruyter, Berlin, 2014. arXiv admin note: substantial text overlap with arXiv:1212.316