Abstract
Dynamical systems possess a computational capacity that may be exploited in a reservoir computing paradigm. This paper presents a general representation of dynamical systems which is based on matrix multiplication. That is similar to how an artificial neural network (ANN) is represented in a deep learning library and its computation can be faster because of the optimized matrix operations that such type of libraries have. Initially, we implement the simplest dynamical system, a cellular automaton. The mathematical fundamentals behind an ANN are maintained, but the weights of the connections and the activation function are adjusted to work as an update rule in the context of cellular automata. The advantages of such implementation are its usage on specialized and optimized deep learning libraries, the capabilities to generalize it to other types of networks and the possibility to evolve cellular automata and other dynamical systems in terms of connectivity, update and learning rules. Our implementation of cellular automata constitutes an initial step towards a more general framework for dynamical systems. Our objective is to evolve such systems to optimize their usage in reservoir computing and to model physical computing substrates. Furthermore, we present promising preliminary results toward the evolution of complex behavior and criticality using genetic algorithm in stochastic elementary cellular automata.
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Notes
- 1.
EvoDynamic open-source repository is available at https://github.com/SocratesNFR/EvoDynamic.
- 2.
Cluster size stands for the number of repetitions of a state that happened consecutively without any interruption of another state.
- 3.
The actual number of histogram bins is normalized or divided by the possible total number of bins.
References
SOCRATES – Self-Organizing Computational substRATES. https://www.ntnu.edu/socrates
Conway’s game of life implemented using tensorflow 2d convolution function (2016). https://github.com/conceptacid/conv2d_life
Aaser, P., et al.: Towards making a cyborg: a closed-loop reservoir-neuro system. In: The 2018 Conference on Artificial Life: A Hybrid of the European Conference on Artificial Life (ECAL) and the International Conference on the Synthesis and Simulation of Living Systems (ALIFE), no. 29, pp. 430–437 (2017). https://doi.org/10.1162/isal_a_072
Abadi, M., et al.: Tensorflow: a system for large-scale machine learning. In: 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–283. USENIX Association, Savannah, GA (2016)
Alstott, J., Bullmore, E., Plenz, D.: Powerlaw: a python package for analysis of heavy-tailed distributions. PLOS ONE 9(1), 1–11 (2014). https://doi.org/10.1371/journal.pone.0085777
Baetens, J.M., Van der Meeren, W., De Baets, B.: On the dynamics of stochastic elementary cellular automata. J. Cell. Automata 12, 63–80 (2016)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987). https://doi.org/10.1103/PhysRevLett.59.381
Broersma, H., Miller, J.F., Nichele, S.: Computational Matter: Evolving Computational Functions in Nanoscale Materials. In: Adamatzky, A. (ed.) Advances in Unconventional Computing. ECC, vol. 23, pp. 397–428. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-33921-4_16
Clauset, A., Shalizi, C.R., Newman, M.E.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)
Gershenson, C.: Introduction to random boolean networks (2004). arXiv preprint nlin/0408006
Goldberg, D.E., Deb, K.: A comparative analysis of selection schemes used in genetic algorithms. In: Foundations of Genetic Algorithms, vol. 1, pp. 69–93. Elsevier (1991)
Goldstein, M.L., Morris, S.A., Yen, G.G.: Problems with fitting to the power-law distribution. Eur. Phys. J. B-Condens. Matter Complex Syst. 41(2), 255–258 (2004)
Jaeger, H., Haas, H.: Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304(5667), 78–80 (2004). https://doi.org/10.1126/science.1091277
Jensen, J.H., Folven, E., Tufte, G.: Computation in artificial spin ice. In: The 2018 Conference on Artificial Life: A Hybrid of the European Conference on Artificial Life (ECAL) and the International Conference on the Synthesis and Simulation of Living Systems (ALIFE), no. 30, pp. 15–22 (2018). https://doi.org/10.1162/isal_a_00011
Kaneko, K.: Overview of coupled map lattices. Chaos: Interdisc. J. Nonlinear Sci. 2(3), 279–282 (1992)
Konkoli, Z., Nichele, S., Dale, M., Stepney, S.: Reservoir Computing with Computational Matter. Comput. Matter. NCS, pp. 269–293. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-65826-1_14
Langton, C.G.: Computation at the edge of chaos: phase transitions and emergent computation. Phys. D: Nonlinear Phenom. 42(1), 12–37 (1990). https://doi.org/10.1016/0167-2789(90)90064-V
Maass, W., Markram, H.: On the computational power of circuits of spiking neurons. J. Comput. Syst. Sci. 69(4), 593–616 (2004). https://doi.org/10.1016/j.jcss.2004.04.001
Nichele, S., Tufte, G.: Trajectories and attractors as specification for the evolution of behaviour in cellular automata. In: IEEE Congress on Evolutionary Computation, pp. 1–8, July 2010. https://doi.org/10.1109/CEC.2010.5586115
Nichele, S., Farstad, S.S., Tufte, G.: Universality of evolved cellular automata in-materio. Int. J. Unconv. Comput. 13(1) (2017)
Nichele, S., Gundersen, M.S.: Reservoir computing using nonuniform binary cellular automata. Complex Syst. 26(3), 225–245 (2017). https://doi.org/10.25088/complexsystems.26.3.225
Nichele, S., Molund, A.: Deep learning with cellular automaton-based reservoir computing. Complex Syst. 26(4), 319–339 (2017). https://doi.org/10.25088/complexsystems.26.4.319
Nichele, S., Tufte, G.: Genome parameters as information to forecast emergent developmental behaviors. In: Durand-Lose, J., Jonoska, N. (eds.) UCNC 2012. LNCS, vol. 7445, pp. 186–197. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32894-7_18
Rendell, P.: Turing Universality of the Game of Life, pp. 513–539. Springer, London (2002). https://doi.org/10.1007/978-1-4471-0129-1_18
Schrauwen, B., Verstraeten, D., Van Campenhout, J.: An overview of reservoir computing: theory, applications and implementations. In: Proceedings of the 15th European Symposium on Artificial Neural Networks 2007, pp. 471–482 (2007)
Subramoney, A., Scherr, F., Maass, W.: Reservoirs learn to learn (2019). arXiv preprint arXiv:1909.07486
Tanaka, G., et al.: Recent advances in physical reservoir computing: a review. Neural Netw. 115, 100–123 (2019). https://doi.org/10.1016/j.neunet.2019.03.005
TensorFlow: tf.sparse.sparse\_dense\_matmul | tensorflow core r1.14 | tensorflow. https://www.tensorflow.org/api_docs/python/tf/sparse/sparse_dense_matmul
Toffoli, T., Margolus, N.: Cellular Automata Machines: A New Environment for Modeling. MIT press, Cambridge (1987)
Wolfram, S.: A New Kind of Science, vol. 5. Wolfram Media, Champaign (2002)
Wright, S.: Correlation and causation. J. Agric. Res. 20, 557–580 (1921)
Acknowledgments
We thank Kristine Heiney for thoughtful discussions about self-organized criticality.
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Pontes-Filho, S. et al. (2020). EvoDynamic: A Framework for the Evolution of Generally Represented Dynamical Systems and Its Application to Criticality. In: Castillo, P.A., Jiménez Laredo, J.L., Fernández de Vega, F. (eds) Applications of Evolutionary Computation. EvoApplications 2020. Lecture Notes in Computer Science(), vol 12104. Springer, Cham. https://doi.org/10.1007/978-3-030-43722-0_9
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