# NORi: An ML-Augmented Ocean Boundary Layer Parameterization

arXiv: https://arxiv.org/abs/2512.04452 · HTML: https://mlsanity.com/paper/2512.04452/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 104 references that are in the map.
- First posted on arXiv 4 Dec 2025.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Inductive Biases for Deep Learning of Higher-Level Cognition](https://mlsanity.com/paper/2011.15091/) (2020)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [Interactive Supercomputing on 40,000 Cores for Machine Learning and Data Analysis](https://mlsanity.com/paper/1807.07814/) (2018)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution](https://mlsanity.com/paper/2304.07029/) (2023)
- [Instead of Rewriting Foreign Code for Machine Learning, Automatically Synthesize Fast Gradients](https://mlsanity.com/paper/2010.01709/) (2020)
- [Learning Subgrid-scale Models with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2212.09967/) (2022)
- [The Ensemble Kalman Inversion Race](https://mlsanity.com/paper/2511.15853/) (2025)

## Built on by (most cited first, 1 of 1)

- [Calibration of a neural network ocean closure for improved mean state and variability](https://mlsanity.com/paper/2604.06398/) (2026)

## Abstract (arXiv)

NORi is a machine learning (ML) parameterization of ocean boundary layer turbulence that is physics-based and augmented with neural networks. NORi stands for neural ordinary differential equations (NODEs) Richardson number (Ri) closure. The physical parameterization is controlled by a Richardson number-dependent diffusivity and viscosity. The neural ODEs are trained to capture the entrainment through the base of the boundary layer, which cannot be represented with a local diffusive closure. The parameterization is trained using large-eddy simulations in an a posteriori fashion, where parameters are calibrated with a loss function that explicitly depends on the actual time-integrated variables of interest rather than the instantaneous subgrid fluxes, which are inherently noisy. NORi conserves tracers by design, uses realistic nonlinear thermodynamics, and demonstrates excellent prediction and generalization capabilities in capturing entrainment dynamics under different convective strengths, background stratifications, rotation, and wind forcings. NORi is shown to simulate the seasonal evolution of the boundary layer at Ocean Weather Station Papa with similar performance to the state-of-the-art two-equation k-epsilon closure. When implemented in a double-gyre simulation, it is numerically stable for at least 100 years, despite only being trained on two-day horizons, and can be run with time steps as long as one hour. Combining highly expressive neural networks with a physically grounded base closure proves to be a robust paradigm for designing parameterizations for climate models: data required and training cost are drastically reduced, inference performance can be directly optimized as a primary objective, and numerical stability is implicitly promoted through training.