Learning vertical coordinates via automatic differentiation of a dynamical core

2025-12-19

AI Summary

Terrain-following coordinates introduce spurious horizontal/vertical motions over steep topography; conventional hybrid or SLEVE coordinates rely on manually tuned heuristic decay functions, compromising geometric fidelity and numerical accuracy. Method: We propose an end-to-end differentiable atmospheric dynamical core, modeling the vertical coordinate as a learnable, monotonically constrained parametric component, and introduce NEUVE—a novel vertical coordinate framework based on invertible integral-transform neural networks. Contribution/Results: NEUVE enables joint gradient-based optimization of coordinate geometry and physical PDE solving, eliminating truncation errors in coordinate derivatives induced by finite-difference approximations. Experiments demonstrate a 1.4–2× reduction in root-mean-square error against nonlinear statistical baselines, complete removal of spurious vertical velocity stripes over steep slopes, and self-adaptive coordinate structuring aligned with both physical constraints and discrete numerical requirements.

Technology Category

• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Develops a learnable vertical coordinate system for atmospheric models
• Reduces spurious motions over steep topography using neural networks
• Optimizes grid structure by coupling simulation errors with parameters

Innovation

• Learned vertical coordinates using neural networks
• Automatic differentiation for exact geometric metric terms
• End-to-end differentiable dynamical core optimization