TheoSea: Marching Theory to Light

There is sufficient information in the far-field of a radiating dipole antenna to rediscover the Maxwell Equations and the wave equations of light, including the speed of light c. TheoSea is a Julia program that does this in about a second, and the key insight is that the compactness of theories drives the search. The program is a computational embodiment of the scientific method: observation, consideration of candidate theories, and validation. 1 Motivation and Background This work flowed 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 recent 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? The initial applications are in electrodynamics (the Maxwell Equations)[2], and there are many other examples such as thermodynamics. Eventually, it is hoped the methods developed will be applicable to data sets from real measurements in a wide variety of fields in physics, engineering, and economics. The Maxwell Equations. The Maxwell Equations in free space with the transformation B′ = cB are: ∇ ·E = 0 (1) ∇ ·B′ = 0 (2) ∇×E + 1 c ∂B′ ∂t = 0 (3) c∇×B′ − ∂E ∂t = 0 (4) ∗Center for Data-Driven Discovery, California Institute of Technology, stalzer at caltech.edu †Minerva Schools at KGI, jj at minerva.kgi.edu 1 ar X iv :1 70 8. 04 92 7v 1 [ cs .A I] 1 4 A ug 2 01 7 where c = 2.99792458×108m/s (MKS units). The spatial-temporal coupling of E and B is how we get electromagnetic waves. The utitility of the B transformation for numerical stability is discussed in Sec. 4 and that is why the Equations look in a slightly strange form in terms of constants. Problem, basic approach, and plan. The problem is to computationally rediscover the Maxwell Equations from data. The data consists of a set of virtual experiments as described in Sec. 2. The experiments are simulated far-field measurements from a dipole antenna. In principle, real data could be used but it is easier to do this purely computationally. This is discussed more in the concluding remarks in Sec. 6. The second step is the generation of candidate theories in Sec. 3, and the third is validation in Sec. 4. Validation is what connects observations to candidate theories and this is the essence of fact based scientific discovery. Here it is done with linear algebra. The final results are in Sec. 5, particularly Fig. 3. But before going into the details, a few comments about past work and Julia. Past work. Attempts to use computers to rediscover physical laws goes back to at least 1979 with BACON.3[4]. The program successfully found the ideal gas law, PV = nRT, from small data tables.1 One of us (Stalzer) and William Xu of Caltech have also rediscovered the ideal gas law with Van der Waals forces using the approach of this paper[5]. In 2009, researchers rediscovered the kinematic equation for the double pendulum essentially using optimization methods to fit constants to candidate equations[6]. What differentiates this work is twofold: the concept of search driven by compactness and completeness, and targeting electrodynamics which is mathematically a much more difficult theory. Indeed, electrodynamics was the first unification (the electric and magnetic fields), and Einstein’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 equation of light as a consequence of the rediscovered free space Maxwell Equations. Julia. TheoSea is written in Julia[7], 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. 2 Observations and the Virtual Experiment The data is from the far-field of a radiating antenna for E,B as shown in the geometry Fig. 1 and data Tab. 1. The fields at a far point P are[8]: E = −00 2 4π ( sin θ r ) cos [ω(t− r/c)]θ̂ (5) B = −00 2 4πc ( sin θ r ) cos [ω(t− r/c)]φ̂ (6) Perhaps the NFL should have consulted BACON.