## 4 Concluding Remarks and Future Work  
There are many avenues for future development as briefly listed below.  
* Expand the enumeration language to allow more expressive theories. Right now _TheoSea_ is limited to theories of the form \(c_{1}A_{1}+c_{2}A_{2}+\ldots\) where the \(A\)'s are operators over fields like \(\mathbf{E},\mathbf{B}\). The Xu ideal gas law code works with scalar fields and exponents. Somehow these should be merged;
* Bigger data and parallelism. The dataset used was very small but semantically very rich. Other datasets, such as for macro-economics, will be far larger. Here Julia's on-the-fly compilation (of candidate theories) and support for parallel processing will be very helpful, and this is one of the reasons the language was chosen (the author encourages the Julia developers to continue work on threads as the model is natural for multicore processors. For example, the main thread could enumerate candidate theories and then send them to several worker threads for validation. At any instant, several theories would be under consideration);  
Figure 4: Average intensity of the Poynting vector, the vector is proportional to \((\mathbf{E}\times\mathbf{B})(\omega)\).  
Figure 3: Screenshot of _TheoSear_’s output showing the Maxwell Equations and the wave equations of light including the speed of light \(c\approx 2.99\times 10^{8}\).  
* Develop methods to work with more noisy data. The current approach works with numerical noise but not measurement noise (Equation (1));
* The fully general Maxwell Equations can be rediscovered just by adding a current \(\mathbf{J}\) and source region \(\rho\). The changes to the virtual experiment and language \(\mathcal{L}\) are straightforward;
* Field discovery. The fields \(\mathbf{E},\mathbf{B}\) are treated as observables. It would be nice if _TheoSea_ could discover the fields from the forces, e.g., \(\mathbf{F}=q\mathbf{E}\). One step is to use a relativistic moving charge \(q\) with velocity \(\mathbf{u}\), where the magnetic field can be written in terms of the electric field [13]: \(\mathbf{B}=(1/c^{2})\,\mathbf{u}\times\mathbf{E}\). Then, the field discovery problem is reduced to finding the electric field and then the magnetic field will fall out from the search.  
However, perhaps the most exciting extension is to apply _TheoSea_ to other domains; we already have results in thermodynamics (entropy) [14] and the Navier-Stokes equations (laminar fluid flow). It is also beginning to look like it should work for the Schrodinger Equation and quantum mechanics [15]. There may also be applications in the social sciences, such as macro-economics. Work is progressing in these areas, and focusing on the applicable representation language and executable semantics are the keys for extensions to new domains.  
The ultimate goal, however, is to generalize _TheoSea_: if a language and validator are supplied, is there a theory? [PERSON] stated [16] that  
_[[PERSON]] knew in advance what the theory should look like, and he explored the theories with mathematical tools, not actual experiments._  
The authors would like to suggest an alternative: _if you know the representation language of a theory and perform a validated search_, you may find the truth. _It may be quicker than you thought._  
The following are available online at [[http://www.mdpi.com/2076-3417/9/14/2899/s1](http://www.mdpi.com/2076-3417/9/14/2899/s1)]([http://www.mdpi.com/2076-3417/9/14/2899/s1](http://www.mdpi.com/2076-3417/9/14/2899/s1)): the Julia code Maxwell.jl.  
All authors contributed equally to this work.  
This research was funded by the Gordon and Betty Moore Foundation through Grant GBMF4915 to the Caltech Center for Data-Driven Discovery.  
Discussions with [PERSON] of Caltech Math/Computer Science were very helpful. The authors are grateful to [PERSON] of Caltech Astronomy and [PERSON] of KGI Natural Sciences for their support.  
The authors declare no conflicts of interest.