## 1 Introduction  
This paper flows from a comment in the concluding remarks of a recent review (2016) of work in data-driven scientific discovery [1]. Specifically,  
_...it may be within current computing and algorithmic technology to infer the Maxwell Equations directly from data given knowledge of vector calculus._  
This paper reports on the progress towards this objective.  
The overarching goal is to develop methods that can infer compact theories from data. Most data intensive analysis techniques are based on machine learning or statistics. These are quite useful, but do not lead to deep understanding or insight. The scientific method and creative scientists have been very good at observations (experiments) and building human understandable models (theory). In this program, we turn both of these ideas on their heads: can a computer given an appropriate _virtual experiment_ (VE) figure out mathematically compact theories [2,3]? The initial application is electrodynamics (the Maxwell Equations) [4]. Eventually, it is hoped the methods developed will be applicable to datasets from real measurements in a wide variety of fields in physics, engineering, and economics.  
**Past work.** Attempts to use computers to rediscover physical laws may have originated in 1979 with BACON.3 [5]. The program successfully found the ideal gas law, \(PV=nRT\), from small data tables. One of out team ([PERSON]) and [PERSON] of Caltech have also rediscovered the ideal gas law with [PERSON] forces using the approach of this paper [6]. In 2009, researchers rediscovered the kinematic equation for the double pendulum essentially using optimization methods to fit constants to candidate equations [7].  
What differentiates this work is twofold: the concept of search driven by compactness and completeness (\"compactness\" here means that we assign weights or complexity scores to each theory being searched; \"completeness\" means that we can always find theories, if any, whose weights are within the specified limit) and targeting a much more difficult theory: electrodynamics. Indeed, electrodynamics was the first unification (the electric and magnetic fields), and [PERSON]'s special relativity is baked right into the equations once the brilliant observation is made that \(c\) is the same in all inertial reference frames. _TheoSea_ also finds the wave equations of light as a consequence of the rediscovered free space Maxwell Equations.  
**Julia.**_TheoSea_ is written in Julia [8], a relatively recent language (roughly 2012) that is both easy to use and has high performance. Julia can be programmed at a high expressive level, and yet, given enough type information, it automatically generates efficient machine code. _TheoSea_ is a Julia _meta-program_ that writes candidate theories in terms of Julia sets that are then validated against data. The set elements are compiled Julia expressions corresponding to terms in the candidate theories. Simultaneously having both high levels of expression and good low-level performance is hard to do in other languages like C and Fortran, and that was a key motivation in Julia's creation.  
**Plan.** The next section presents some preliminaries that are used by _TheoSea_ in general. Central to the approach is the section on rapid enumeration (Section 2.1). Section 3 then presents the virtual experiment, rediscovery, and run-times. The paper ends with some concluding remarks (Section 4).