Abstract. This paper proposes a novel design of the Sigma‑Pi neuron with the controlled automatic activation of the factor multiplication operation during the training phase of a neural network. The solution can be transparently integrated into both current and prospective neural architectures; it performs direct neural computation of a wide range of mathematical relations that are commonly implemented by neural networks in a memoization mode. The direct execution of the multiplication operation yields smaller network errors; this advantage is very clear when the network processes input data outside the training domain. As proved below, a single Sigma‑Pi neuron can implement the XOR function of arbitrary dimension. The effectiveness and computational capabilities of the novel design, as well as its superiority over classical fully connected feedforward neural networks, are demonstrated for the neural modeling of polynomials and operations of vector‑matrix algebra. The proposed Sigma‑Pi neuron design is expected to find application both in surrogate neural network models for computation and simulation and in general AI+Science tasks.
Keywords: Sigma‑Pi neuron, factor multiplication operation, XOR function, surrogate models.
Acknowledgments. The authors are grateful to Academician D.A. Novikov for his permanent attention to this work, to D.S. Gadzhiev for his active discussion of the related issues and valuable remarks, and to V.V. Kazakov for his implementation of the neuron in the PyTorch framework.
Portsev, R.Yu. and Makarenko, A.V., The Sigma-Pi Neuron and Resolving the General “Factor Multiplication” Problem in Artificial Neural Networks. Control Sciences 4, 12–23 (2026).
The Sigma-Pi Neuron and Resolving the General “Factor Multiplication” Problem in Artificial Neural Networks