AI 中文总结
提出两阶段神经框架，通过物理信息模型生成外力轨迹并优化，结合数学认证准则，证明三维不可压缩Navier-Stokes方程有限时间爆破。
AI 中文摘要
我们提出一个用于受迫三维不可压缩Navier-Stokes流动的两部分神经框架。第一部分开发计算强迫系统。一个物理信息神经模型生成结构化的外力轨迹，候选解通过可微PDE滚动或PPO-Clip进行优化，选定的强迫被冻结并通过独立的固定强迫重放进行验证。第二部分提供数学认证层。它将神经候选发现与连续介质分析分离，推导出积分倒数涡量准则，该准则蕴含Riccati型增长和光滑延拓的有限时间丧失，开发了一种经过验证的计算到连续介质的转移策略，并为非退化神经输出律建立了条件正概率闭包。证明在连续介质层面是完整的。
英文摘要
We present a two-part neural framework for forced three-dimensional incompressible Navier--Stokes flow. Part~I develops the computational forcing system. A physics-informed neural model generates structured external-force trajectories, candidates are optimized through differentiable PDE rollouts or PPO-Clip, and selected forcings are frozen and checked by independent fixed-force replay. Part~II provides the mathematical certification layer. It separates neural candidate discovery from continuum analysis, derives integrated reciprocal-vorticity criteria that imply Riccati-type growth and finite-time loss of smooth continuation, develops a validated computational-to-continuum transfer strategy, and establishes a conditional positive-probability closure for a nondegenerate neural output law. The proof is complete at the continuum level.
