AdS/CFT String Theory Boltzmann Machine Python Applications and Future Directions

This is the final articles in the series that will bring together the concepts from the previous ones, plus a Python Machine Learning-AI demonstration. With the previous articles on the Deep Boltzmann Machine (DBM) AdS-CFT analogies in hand, we can now demonstrate a simple Boltzmann Machine application. We have alluded to some large applications in the previous sections [25][26]. Now we will focus on a simple Machine Learning and Artificial Intelligence (AI) example using a familiar dataset.

One possible future direction for this research is to incorporate the Schr?dinger wave equation, the primary quantum wave equation and ubiquitous in particle physics [22][23][24]. It has been used extensively in other Ising models with wave particle dualities.

The Schr?dinger wave equation is defined in our previous papers as crucial to certain conformal maps for defining the time evolving space structure as a function of quantum field theory. This again is an effort to correlate gravity and quantum mechanics in the AdS-CFT correspondence. The correlation is done in Hilbert Space, which is the primary working space for quantum mechanics, defined as the inner product or overlap of two vectors [5][16][18].

The analytical results of the AdS Boltzmann Machine model were testing using the Python boltzmannclean Library https://github.com/facultyai/boltzmannclean [27].

This library uses a Restricted Boltzmann Machine (RBM) to train and sample the data then clean and fill in and clean missing values from a Pandas Data Frame. The data can either be numerical or categorical. Hyper parameters include the following:

???????????n_hidden: the size of the hidden layer

???????????learn_rate: learning rate for stochastic gradient descent, or the most efficient path to optimize the learning rate. This is a slope calculation to determine the steepest slope to the desired optimization value. In this case it is stochastic so the slope is nonlinear.

???????????batchsize: batchsize for stochastic gradient descent

???????????dropout_fraction: fraction of hidden nodes to be dropped out on each backward pass during training

???????????max_epochs: maximum number of passes over the training data

???????????adagrad: whether to use the Adagrad update rules for stochastic gradient descent

We used the well-known Iris dataset to test, which is the same dataset used in the GitHub developer site. We wanted to verify that our model was working correctly and produced consistent results with the original. Here are the results in Python Jupyter Notebook. Here we import the necessary libraries, then read in the dataset:

Here is where we add the noise or empty values to the data using (row,col):

Here are the noisy missing NaN values:

Finally we use boltzmann.clean to clean up the noise values with the DBM predicted values.?

Here are the results:

Compared to the original dataset: we can see where the DBM predicted the 4.9,4.7 values as 6.3, and the 1.4 values as 4.4. This is not as accurate as we hoped, but with a larger dataset we can further train the model to be more accurate and have a lower error value.?

References

[1] Eugeny Babichev, Philippe Brax,Chiara Caprini, J′er?ome Martin, Dani`ele A. Steer. Dirac Born Infeld (DBI) Cosmic Strings. 11 Sep 2008. arXiv:0809.2013v1 [hep-th].

[2] K. Becker, M. Becker, J. & Schwartz. String Theory and M-Theory: An Introduction. Cambridge University Press, New York. ISBN: 10-521-86069-5. 2007.

[3] N. Beisert et al., \Review of AdS/CFT Integrability: An Overview", Lett. Math. Phys. 99, 3 (2012), arXiv:112.3982.

[4] Brodsky, Stanley. (2008). Novel LHC Phenomena. 002. 10.22323/1.045.0002.

[5] B. Duplantier et al. Schramm Loewner Evolution and Liouville Quantum Gravity. Phys.Rev.Lett. 107 (2011) 131305 arXiv:1012.4800 [math-ph].

[6] Razvan Ciuca, Oscar F. Hern′andez and Michael Wolman. A Convolutional Neural Network For Cosmic String Detection in CMB Temperature Maps. 14 Mar 2019.arXiv:1708.08878v3 [astro-ph.CO]

[7] Et. Al. Constraints on cosmic strings using data from the ?rst Advanced LIGO observing run. [Submitted on 11 Sep 2008]https://arxiv.org/abs/1712.01168v2

[8] Isabel Fernandez-Nu~nez and Oleg Bulashenko. Wave propagation in metamaterials mimicking the topology of a cosmic string. 6 Mar 2018. arXiv:1711.02420v2 [physics.optics].

[9] O. Gonz′alez-Gaxiola and J. A. Santiago. Symmetries, Mellin Transform and the Black-Scholes Equation (A Nonlinear Case). Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 10, 469 – 478. HIKARI Ltd, www.m-hikari.com. https://dx.doi.org/10.12988/ijcms.2014.4673

[10] Koji Hashimoto. AdS/CFT as a deep Boltzmann machine. Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan. (Dated: March 13, 2019). https://arxiv.org/abs/1903.04951

[11] Junkee Jeon and Ji-Hun Yoon. Discount Barrier Option Pricing with a Stochastic Interest Rate: Mellin Transform Techniques and Method of Images. Commun. Korean Math. Soc. 33 (2018), No. 1, pp. 345–360. https://doi.org/10.4134/CKMS.c170060. pISSN: 1225-1763 / eISSN: 2234-3024.

[12] E. Kiritsis. String Theory in a Nutshell. Princeton University Press. ISBN: 10:-0-691-12230-X. 19 March 2007.

[13] Vihar Kurama. Beginner's Guide to Boltzmann Machines in PyTorch. May 2021. https://blog.paperspace.com/beginners-guide-to-boltzmann-machines-pytorch/

[14] M. Li, R. Miao & R. Zheng. Meta-Materials Mimicking Dynamic Spacetime D-Brane and Non-Commutativity in String Theory. 03 February 2011. arXiv: 1005.5585v2..

[15] S. Little. AdS-CFT Stochastic Feynman-Kac Mellin Transform with Chaotic Boundaries. Academia.edu. December 28, 2021. https://www.academia.edu/66244508/AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform_with_Chaotic_Boundaries?source=swp_share.

[16] S. Little. Chaotic Boundaries of AdS-CFT Stochastic Feynman-Kac Mellin Transform. Academia.edu. December 28, 2021. https://www.academia.edu/66245245/Chaotic_Boundaries_of_AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform?source=swp_share.

[17] S. Little. Feynman-Kac Formulation of Stochastic String DBI Helmholtz Action. Academia.edu. July 13, 2021. https://www.academia.edu/49860679/Feynman_Kac_Formulation_of_Stochastic_String_DBI_Helmholtz_Action.

[18] S. Little. Liouville SLE Boundaries on CFT Torus Defined with Stochastic Schr?dinger Equation. SIAM Conference on Analysis of Partial Differential Equations (PD11) December 7-10, 2015.?https://www.siam.org/meetings/pd15/. Session:?https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=2189311

[19] S. Little. Session Chair and Contributed Speaker (CP10): Stochastic Helmholtz Finite Volume Method for DBI String-Brane Theory Simulations. SIAM Conference on Analysis of Partial Differential Equations (PD19),?December 11 – 14, 2019, La Quinta Resort & Club, La Quinta, California, Session:?https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=67694

[20] David A. Lowe. Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature. 17 Feb 2016. arXiv:1602.05613v1 [hep-th]

[21] J. M. Maldacena, “The large N limit of superconformal ?eld theories and super-gravity,” Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].

[22] Pallab Basu, Diptarka Das, Leopoldo A. Pando-Zayas, Dori Reichmann. Chaos in String Theory.[arXiv:1103.4101, 1105.2540, 1201.5634, and ongoing work.]

[23] Pallab Basu, Leopoldo A. Pando Zayas, Phys.Lett.B 11 (2011) 00418, [arXiv:1103.4107]. Pallab Basu, Diptarka Das, Archisman Ghosh, Phys.Lett.B 11 (2011) 00417, [arXiv:1103.4101].

[24] S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. 23 Sep 2015. arXiv:1012.4797v2 [math.PR]

[25] UCLA Ozcan Research Group. UCLA engineers use deep learning to reconstruct holograms and improve optical microscopy. November 20, 2017. https://phys.org/news/2017-11-ucla-deep-reconstruct-holograms-optical.html

[26] Favio Vázquez. Deep Learning made easy with Deep Cognition. Dec 21, 2017. https://becominghuman.ai/deep-learning-made-easy-with-deep-cognition-403fbe445351

[27] Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.