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This research establishes rigorous mathematical conditions for neural turbulence closures, proving that enforcing global tangent dissipation ensures solution uniqueness and significantly reduces sensitivity to residuals in high-Reynolds-number flows. By combining exact-integral constructions with penalty methods, the study demonstrates how learned models can maintain stability without degrading the underlying physics of diffusion and reaction terms. These findings advance the theoretical foundation for applying machine learning to complex fluid dynamics problems where traditional numerical methods face convergence challenges.
Well-posedness of neural turbulence closures and tangent dissipation
arXiv:2609.19647v1 Announce Type: cross Abstract: 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 $Re_\tau=180$--$5200$, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At $Re_\tau=1000$, 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 $Re_\tau=5200$, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately $372\times$ and $11\times$, respectively.
Abstract: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 $Re_\tau=180$--$5200$, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At $Re_\tau=1000$, 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 $Re_\tau=5200$, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately $372\times$ and $11\times$, respectively.
Fluid Dynamics (physics.flu-dyn); Machine Learning (cs.LG)
arXiv:2609.19647 [physics.flu-dyn]
(or arXiv:2609.19647v1 [physics.flu-dyn] for this version)
https://doi.org/10.48550/arXiv.2609.19647 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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