Optimizing and certifying the positivity of polynomials are fundamental primitives across mathematics and engineering applications, from dynamical systems to operations research. However, solving these problems in practice requires large semidefinite programs, with poor scaling in dimension and degree. In this work, we demonstrate for the first time that neural networks can effectively solve such problems in a data-driven fashion, achieving tenfold speedups while retaining high accuracy. Moreover, we observe that these polynomial learning problems are equivariant to the non-compact group $SL(2,\mathbb{R})$, which consists of area-preserving linear transformations. We therefore adapt our learning pipelines to accommodate this structure, including data augmentation, a new $SL(2,\mathbb{R})$-equivariant architecture, and an architecture equivariant with respect to its maximal compact subgroup, $SO(2, \mathbb{R})$. Surprisingly, the most successful approaches in practice do not enforce equivariance to the entire group, which we prove arises from an unusual lack of architecture universality for $SL(2,\mathbb{R})$ in particular. A consequence of this result, which is of independent interest, is that there exists an equivariant function for which there is no sequence of equivariant polynomials multiplied by arbitrary invariants that approximates the original function. This is a rare example of a symmetric problem where data augmentation outperforms a fully equivariant architecture, and provides interesting lessons in both theory and practice for other problems with non-compact symmetries.

翻译：多项式正性优化与验证是数学及工程应用（从动力系统到运筹学）中的基本工具。然而，实际求解这类问题需要处理大规模半定规划，其复杂度随维度和阶数急剧增长。本文首次证明，神经网络能以数据驱动方式高效求解此类问题，在保持高精度的同时实现十倍加速。此外，我们发现多项式学习问题对非紧群$SL(2,\mathbb{R})$（由保面积线性变换构成）具有等变性。为此，我们设计了适配该结构的框架，包括数据增强、新型$SL(2,\mathbb{R})$等变架构，以及针对其最大紧子群$SO(2,\mathbb{R})$的等变架构。令人惊讶的是，实践中表现最优的方法并未强制执行全群等变性——我们证明这是由于$SL(2,\mathbb{R})$存在特殊的架构通用性缺失。这一独立意义的结论表明：存在某个等变函数，无法被任意不变量乘以等变多项式序列所逼近。这是非紧对称问题中数据增强超越完全等变架构的罕见案例，为非紧对称性问题的理论与实用化提供了重要启示。