发表机构
Netherlands Defence Academy; Data Science Centre of Excellence, NL Ministry of Defence（荷兰国防学院; 荷兰国防部数据科学卓越中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出NeuralCert框架，将神经网络学习的高维变分试验函数经谱诊断、剪枝与多模求值精确认证，在三个极值问题上展示了神经优化可发现改进构造、揭示经验不变量及优化障碍，为AI辅助严格数学提供路径。
AI 中文摘要
神经网络在解决数学问题方面日益流行，但随机模型本身并不提供数学上的精确性。本研究引入了一个从发现到认证的框架，其中高维变分试验函数以紧凑的可分离表示进行学习，经过谱诊断和剪枝，然后通过多模求值进行精确认证。精确认证使数值证明完全显式且可独立验证。该框架可在标准个人计算机上运行。在三个极值问题中，我们展示了神经优化能够以三种不同方式为严格数学做出贡献：通过发现改进的构造，通过揭示导致证明的经验不变量，以及通过揭示其几何结构激发新的解析或数值表示的优化障碍。更广泛地，这些结果提示了一条通往AI辅助数学的路径，其中灵活的计算发现和精确认证成为单一严格工作流程中的互补组成部分。
英文摘要
Neural networks are becoming popular in solving mathematical problems, but stochastic models do not provide mathematical exactness by themselves. This study introduces a discovery-to-certification framework in which high-dimensional variational trial functions are learned in a compact separable representation, spectrally diagnosed and pruned, and then certified exactly through multimodular evaluation. Exact certification makes the numerical proofs fully explicit and independently verifiable. This framework can be run on a standard personal computer. Across three extremal problems, we show that neural optimization can contribute to rigorous mathematics in three distinct ways: by discovering improved constructions, by exposing empirical invariants that lead to proofs, and by revealing optimization barriers whose geometry motivates new analytic or numerical representations. More broadly, these results suggest a path toward AI-assisted mathematics in which flexible computational discovery and exact certification become complementary components of a single rigorous workflow.
Comments86 pages, 3 Figures
