Well-posedness of neural turbulence closures and tangent dissipation
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 Reτ=180--$5200$, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At Reτ=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 Reτ=5200, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately 372× and 11×, respectively.
Related
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- Generalizable turbulence closures across bluff-body shapes by PINN-based solver-agnostic training
- Fluctuation-dissipation relations in isotropic turbulence from Kraichnan's fully resolved DIA closure
- Scaling laws and energy dissipation in the dynamics of SLE curve as a 1D turbulence model
- Scalar Dissipation Criticality in Compressible Magnetized Turbulence
BibTeX
@misc{zhang2026wellposedness,
title = {{Well-posedness of neural turbulence closures and tangent dissipation}},
author = {Zhen Zhang and George Em Karniadakis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19647},
eprint = {2609.19647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19647},
}
RIS
TY - UNPB
TI - Well-posedness of neural turbulence closures and tangent dissipation
AU - Zhen Zhang
AU - George Em Karniadakis
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19647
UR - https://arxiv.org/abs/2609.19647
AB - 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× and 11×, respectively.
ER -