发表机构
Brown University（布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对神经湍流闭合，证明全局切向耗散保证单调算子的适定性，并提出两种促进方法，在槽道流测试中验证了唯一性并大幅降低逆灵敏度。
AI 中文摘要
神经湍流闭合定义了一个新的边值问题，$R(U)=N(U)+F(U)=0$，其耦合雅可比矩阵为 $J(U)=N'(U)+F'(U)$，其中 $N$ 是原始平均流算子，$F$ 是学习得到的闭合项。我们建立了全局切向耗散的两个推论。对于单调原始算子，由原始算子和闭合项共同提供的正均匀余量保证了存在性、唯一性以及将后验解误差与先验残差联系起来的全局逆灵敏度界。对于一般原始算子，耗散性闭合不能恶化切向耗散，但仅此并不保证唯一性。切向耗散依赖于扩散和反应两者。我们研究了两种互补的促进方法：（1）一种精确积分构造，强制非负切向扩散，同时不约束反应；（2）对采样状态下的切向反应违规施加惩罚。切向扩散进入雅可比矩阵，仅非负割线涡粘性并不能控制其强制性。我们在 $Re_\ au=180$--$5200$ 的槽道流上进行了测试，这提供了一个强单调基线。两种受约束的闭合在所有50个训练种子/雷诺数案例中均达到了精确解。在 $Re_\ au=1000$ 时，我们对每种闭合的一个固定网络进行了10000次启动测试，发现每种受约束闭合有一个根，而其他闭合有多个根。虽然这不能证明唯一性，但为所测试的受约束闭合的唯一性提供了强有力的经验证据。在 $Re_\ au=5200$ 时，构造和惩罚分别将报告的相对于原始算子的逆灵敏度降低了约 $372\ imes$ 和 $11\ imes$。
英文摘要
A neural turbulence closure defines a new boundary-value problem, $R(U)=N(U)+F(U)=0$, with a coupled Jacobian $J(U)=N'(U)+F'(U)$, where $N$ is the original mean-flow operator and $F$ the learned closure. We establish two consequences of global tangent dissipation. For a monotone original operator, a positive uniform margin supplied by the original operator and closure together guarantees existence, uniqueness and a global inverse-sensitivity bound relating a posteriori solution error to the a priori residual. For a general original operator, a dissipative closure cannot worsen tangent dissipation, but this alone does not guarantee uniqueness. Tangent dissipation depends on both diffusion and reaction. We study two complementary ways to promote it: (1) an exact-integral construction enforcing non-negative tangent diffusion while leaving reaction unconstrained, and (2) a penalty on tangent-reaction violations at sampled states. Tangent diffusion enters the Jacobian, and non-negative secant eddy viscosity alone does not control its coercivity. We conduct tests with channel flow at $Re_τ=180$--$5200$, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At $Re_τ=1000$, we conduct tests with 10,000 starts for one fixed network per closure and we find one root for each constrained closure and multiple roots for the other closures. Although this does not prove uniqueness, it provides strong empirical evidence for uniqueness of the tested constrained closures. At $Re_τ=5200$, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately $372\times$ and $11\times$, respectively.
Comments20 pages, 4 figures
