arXiv subjects

Seth Taylor

Publications and source records attributed to Seth Taylor.

5 recordsLinked to original sources

arXiv2026

Thermodynamically Constrained Information Geometric Regularization for Compressible Flows

We construct and analyze a thermodynamic extension of the recently proposed information geometric regularization of Cao and Sch\"afer. The construction extends their shock-mitigating Hessian metric geometry using the Shannon entropy to constrain the regularized motion based on a thermodynamic length. Reformulating the equations in terms of mass and specific entropy explicitly connects the thermodynamic state to a position in the diffeomorphism group, allowing for a derivation of the regularized equations using an information geometric mechanics formalism based on geodesics on a Hessian manifold with a dual affine connection. The dynamics are defined using a pullback geometry for the Levi--Civita connection, describing constrained geodesic motion, and the cubic Amari--Chentsov tensor describing the information geometric correction. This new compressible fluid model introduces an anisotropic stress tensor to the momentum equation that vanishes along isentropic directions and an additional elliptic equation coupled to the barotropic regularization. Numerical simulations in one and two spatial dimensions demonstrate that the geometrically consistent incorporation of a thermodynamic constraint mitigates cusp singularities previously observed in other approaches while still maintaining the benefits of an inviscid regularization.

math.NA↗

arXiv2025

Learning vertical coordinates via automatic differentiation of a dynamical core

Terrain-following coordinates in atmospheric models often imprint their grid structure onto the solution, particularly over steep topography, where distorted coordinate layers can generate spurious horizontal and vertical motion. Standard formulations, such as hybrid or SLEVE coordinates, mitigate these errors by using analytic decay functions controlled by heuristic scale parameters that are typically tuned by hand and fixed a priori. In this work, we propose a framework to define a parametric vertical coordinate system as a learnable component within a differentiable dynamical core. We develop an end-to-end differentiable numerical solver for the two-dimensional non-hydrostatic Euler equations on an Arakawa C-grid, and introduce a NEUral Vertical Enhancement (NEUVE) terrain-following coordinate based on an integral transformed neural network that guarantees monotonicity. A key feature of our approach is the use of automatic differentiation to compute exact geometric metric terms, thereby eliminating truncation errors associated with finite-difference coordinate derivatives. By coupling simulation errors through the time integration to the parameterization, our formulation finds a grid structure optimized for both the underlying physics and numerics. Using several standard tests, we demonstrate that these learned coordinates reduce the mean squared error by a factor of 1.4 to 2 in non-linear statistical benchmarks, and eliminate spurious vertical velocity striations over steep topography.

physics.ao-ph↗

arXiv2025

Diffeomorphic Neural Operator Learning

We present an operator learning approach for a class of evolution operators using a composition of a learned lift into the space of diffeomorphisms of the domain and the group action on the field space. In turn, this transforms the semigroup structure of the evolution operator into a corresponding group structure allowing time stepping be performed through composition on the space of diffeomorphisms rather than in the field space directly. This results in a number of structure-preserving properties related to preserving a relabelling symmetry of the dynamics as a hard constraint. We study the resolution properties of our approach, along with its connection to the techniques of diffeomorphic image registration. Numerical experiments on forecasting turbulent fluid dynamics are provided, demonstrating its conservative properties, non-diffusivity, and ability to capture anticipated statistical scaling relations at sub-grid scales. Our method provides an example of geometric operator learning and indicates a clear performance benefit from leveraging a priori known infinite-dimensional geometric structure.

math.NA↗

arXiv2023

A characteristic mapping method for incompressible hydrodynamics on a rotating sphere

We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as a composition of sub-interval flows formed by $C^1$ spherical spline interpolants. This approximation technique has the capacity of resolving sub-grid scales generated over time without increasing the spatial resolution of the computational grid. The numerical method is analyzed and validated using standard test cases yielding third-order accuracy in the supremum norm. Numerical experiments illustrating the unique resolution properties of the method are performed and demonstrate the ability to reproduce the forward energy cascade at sub-grid scales by upsampling the numerical solution.

math.NA↗

arXiv2021

A Characteristic Mapping Method for Tracer Transport on the Sphere

A semi-Lagrangian Characteristic Mapping method for the solution of the tracer transport equations on the sphere is presented. The method solves for the solution operator of the equations by approximating the inverse of the diffeomorphism generated by a given velocity field. The evolution of any tracer and mass density can then be computed via pullback with this map. We present a spatial discretization of the manifold-valued map using a projection-based approach with spherical spline interpolation. The numerical scheme yields $C^1$ continuity for the map and global second-order accuracy for the solution of the tracer transport equations. Error estimates are provided and supported by convergence tests involving solid body rotation, moving vortices, deformational, and compressible flows. Additionally, we illustrate some features of computing the solution operator using a numerical mixing test and the transport of a fractal set in a complex flow environment.

math.NA↗