Fuzzy Systems with Sigmoid-Based Membership Functions as Interpretable Neural Networks

  • Barnabás BedeEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)


In this paper new interpretable neural network architectures are proposed. A Neural Network with sigmoid activation function is converted into a sigmoid-based approximation operator, which, at its turn, can be approximated by a fuzzy system of Takagi-Sugeno type. Altogether this process shows that a neural network with sigmoid activation functions can be approximated by a Takagi-Sugeno fuzzy system. As the TS fuzzy system provides interpretability, while Neural Networks provide approximation capability, we obtain a novel interpretable Neural Network Architecture. Interpretability of the TS fuzzy system can provide insights into data in a new way, enhancing decision making.


  1. 1.
    Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Bonanno, D., Nock, K., Smith, L., Elmore, P., Petry, F.: An approach to explainable deep learning using fuzzy inference. In: Next-Generation Analyst V, SPIE Proceedings, vol. 10207, International Society for Optics and Photonics (2017)Google Scholar
  3. 3.
    Costarelli, D., Spigler, R.: Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101–106 (2013)CrossRefGoogle Scholar
  4. 4.
    Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2(4), 303–314 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Horikawa, S.-I., Furuhashi, T., Uchikawa, Y.: On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm. IEEE Trans. Neural Netw. 3(5), 801–806 (1992)CrossRefGoogle Scholar
  6. 6.
    Jang, J.-S.R.: ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans. Syst. Man Cybern. 23(3), 665–685 (1993)CrossRefGoogle Scholar
  7. 7.
    Jin, Y., Sendhoff, B.: Extracting interpretable fuzzy rules from RBF networks. Neural Process. Lett. 17(2), 149–164 (2003)CrossRefGoogle Scholar
  8. 8.
    Kasabov, N.K.: Learning fuzzy rules and approximate reasoning in fuzzy neural networks and hybrid systems. Fuzzy Sets Syst. 82(2), 135–149 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Montavon, G., Samek, W., Muller, K.R.: Methods for interpreting and understanding deep neural networks. Digit. Signal Process. 73, 1–15 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sugeno, M.: An introductory survey of fuzzy control. Inf. Sci. 36, 59–83 (1985)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 45(2), 135–156 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)CrossRefGoogle Scholar
  13. 13.
    Ying, H.: General SISO Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators. IEEE Trans. Fuzzy Syst. 6(4), 582–587 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zadeh, L.A.: Fuzzy Sets. Inf. Control 8, 338–353 (1965)CrossRefGoogle Scholar
  15. 15.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - I. Inf. Sci. 8(3), 199–249 (1975)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zadeh, L.A.: Fuzzy logic, neural networks, and soft computing. Fuzzy Sets, Fuzzy Logic, And Fuzzy Systems, pp. 775–782 (1996). Selected Papers by Zadeh, L.AGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.DigiPen Institute of TechnologyRedmondUSA

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