Early-career research(J-PEAKS)

Algebraic equation systems appear in a wide range of applications, including large-scale algebraic computations, dynamical system analysis, computer vision, and post-quantum cryptography. Unlike numerical methods, algebraic approaches provide exact solutions, offering deep insights into system evolution, response characteristics, and stability. However, the computation of Gröbner bases, which forms the foundation for many algebraic methods, suffers from exponential computational complexity with respect to the number of variables. This project aims to address this challenge through a novel deep learning approach, specifically targeting the dramatic acceleration of Gröbner basis computation, which plays a central role in these methods. Task 1 focuses on the algebra of dataset construction. Learning requires pairs of algebraic equation systems and their Gröbner bases (F,G). To generate these, we obtain a suitable Gröbner basis G and transform it into an equivalent algebraic equation system F. This reverse approach is unexplored in computational algebra. We aim to identify classes of algebraic equation systems that can efficiently generate learning samples and develop algorithms for their generation. Task 2 addresses the learning of input-sensitive functions.

Algebraic computation learning requires functions that are sensitive to changes in input coefficients or operation symbols. To address this, we will redesign deep learning models for input-sensitive function learning and establish their theoretical foundations. The inability of algebraic equation system computations to scale with the number of variables, hindering their application to real-world systems, has been a long-standing challenge. Solving Tasks 1 and 2 will have a profound impact not only in the field of system control but also in a wide range of fields involving algebraic and differential equation systems. This research concept pioneers new areas of "computational algebra for learning" and "deep learning for computation" through the novel approach of "learning".

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