Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917400835784704 |
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| author | Yoneda, Tsuyoshi |
| author_facet | Yoneda, Tsuyoshi |
| contents | We consider the incompressible three-dimensional Euler equations for a vortex ring with Kelvin waves undergoing radially expanding Lagrangian transport. To clarify the fundamental mechanisms underlying nonlinear scale-local deformations of the vortex structure, we develop a geometric Lagrangian framework that avoids singular integral representations of the pressure and yields a novel wave equation governing the axis of swirling particles. Within this framework, we identify intrinsic nonlinear mechanisms that drive scale-local deformations of the vortex structure, supported by a machine-learning-based analysis. Specifically, we propose a hybrid optimization framework that combines Bayesian global exploration with adjoint-based local refinement. The resulting optimization problem exhibits a highly non-convex loss landscape, in which the adjoint method alone fails to escape local minima. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods Yoneda, Tsuyoshi Analysis of PDEs We consider the incompressible three-dimensional Euler equations for a vortex ring with Kelvin waves undergoing radially expanding Lagrangian transport. To clarify the fundamental mechanisms underlying nonlinear scale-local deformations of the vortex structure, we develop a geometric Lagrangian framework that avoids singular integral representations of the pressure and yields a novel wave equation governing the axis of swirling particles. Within this framework, we identify intrinsic nonlinear mechanisms that drive scale-local deformations of the vortex structure, supported by a machine-learning-based analysis. Specifically, we propose a hybrid optimization framework that combines Bayesian global exploration with adjoint-based local refinement. The resulting optimization problem exhibits a highly non-convex loss landscape, in which the adjoint method alone fails to escape local minima. |
| title | Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.22543 |