基于Transformer的神经形状函数求和：用于多项式非负性认证

多项式非负性认证是已知的NP难问题，具有非凸优化、控制、机器人等领域的直接应用。非负性的充分条件是形状函数求和（Sum of Squares, SOS）属性，即可表示为其他多项式的平方和。在实际中， however, certifying the SOS criterion remains computationally expensive and often involves solving a Semidefinite Program (SDP), whose dimensionality grows quadratically in the size of the monomial basis of the SOS expression; hence, various methods to reduce the size of the monomial basis have been proposed. In this work, we introduce the first learning-augmented algorithm to certify the SOS criterion. To this end, we train a Transformer model that predicts an almost-minimal monomial basis for a given polynomial, thereby drastically reducing the size of the corresponding SDP. Our overall methodology comprises three key components: efficient training dataset generation of over 100 million SOS polynomials, design and training of the corresponding Transformer architecture, and a systematic fallback mechanism to ensure correct termination, which we analyze theoretically. We validate our approach on over 200 benchmark datasets, achieving speedups of over $100\times$ compared to state-of-the-art solvers and enabling the solution of instances where competing approaches fail. Our findings provide novel insights towards transforming the practical scalability of SOS programming.