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Gaussian processes at the Helm(holtz): A more fluid model for ocean currents

Renato Berlinghieri, Brian L. Trippe, David R. Burt, Ryan Giordano, Kaushik Srinivasan, Tamay Özgökmen, Junfei Xia, Tamara Broderick

TL;DR

This work tackles reconstructing 2D ocean currents and their divergences from sparse buoy data by challenging the common velocity GP approach. It introduces a Helmholtz Gaussian process that places independent GPs on the divergent and rotational components via potentials $\Phi$ and $\Psi$, yielding a current field $F = \mathrm{grad}\,\Phi + \mathrm{rot}\,\Psi$ and enabling direct inference of divergence $\delta$ and vorticity $\zeta$. The resulting $k_{\mathrm{Helm}}$ kernel preserves tractable GP inference while better aligning priors with fluid-dynamics, including frame equivariance and differing scales for divergence and vorticity. Across simulations and real drifter data (LASER and GLAD), the Helmholtz GP yields more accurate current reconstructions and more physically plausible divergence/vorticity patterns, with computational costs comparable to standard velocity GPs. This approach provides a practical, physics-informed framework for vector-field inference from sparse observations in oceanography.

Abstract

Given sparse observations of buoy velocities, oceanographers are interested in reconstructing ocean currents away from the buoys and identifying divergences in a current vector field. As a first and modular step, we focus on the time-stationary case - for instance, by restricting to short time periods. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current reconstruction and divergence identification, due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method with theory and experiments on synthetic and real ocean data.

Gaussian processes at the Helm(holtz): A more fluid model for ocean currents

TL;DR

This work tackles reconstructing 2D ocean currents and their divergences from sparse buoy data by challenging the common velocity GP approach. It introduces a Helmholtz Gaussian process that places independent GPs on the divergent and rotational components via potentials and , yielding a current field and enabling direct inference of divergence and vorticity . The resulting kernel preserves tractable GP inference while better aligning priors with fluid-dynamics, including frame equivariance and differing scales for divergence and vorticity. Across simulations and real drifter data (LASER and GLAD), the Helmholtz GP yields more accurate current reconstructions and more physically plausible divergence/vorticity patterns, with computational costs comparable to standard velocity GPs. This approach provides a practical, physics-informed framework for vector-field inference from sparse observations in oceanography.

Abstract

Given sparse observations of buoy velocities, oceanographers are interested in reconstructing ocean currents away from the buoys and identifying divergences in a current vector field. As a first and modular step, we focus on the time-stationary case - for instance, by restricting to short time periods. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current reconstruction and divergence identification, due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method with theory and experiments on synthetic and real ocean data.
Paper Structure (54 sections, 15 theorems, 71 equations, 4 figures, 2 tables)

This paper contains 54 sections, 15 theorems, 71 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian Processes at the Helm(holtz)
  4. Advantages of the Helmholtz prior
  5. Experimental Results
  6. Simulated experiments
  7. Real-data experiments
  8. Discussion and Future Work
  9. Appendix
  10. Related Work
  11. Velocity GP vs. the GP of goncalves2019naivelodise2020investigating.
  12. Why we focused on the Helmholtz decomposition.
  13. Divergence, Gradient, and Curl Operators in 2D
  14. Helmholtz Decomposition in the Ocean
  15. Optimal Interpolation vs. Gaussian Processes
  16. ...and 39 more sections

Key Result

Proposition 3.0

Let $F$ be an ocean current vector field defined by potential and stream functions that are independent and distributed as $\Phi \sim \mathcal{GP}(0, k_{\Phi})$ and $\Psi \sim \mathcal{GP}(0, k_{\Psi})$, where $k_{\Phi}$ and $k_{\Psi}$ are such that $\Phi$ and $\Psi$ have almost surely continuously where, for $\mathbf{x}, \mathbf{x}'\in \mathbb{R}^2$, $i,j \in {1,2}$, $k_{\mathrm{Helm}}(\mathbf{x

Figures (4)

  • Figure 1: First column: ground truth predictions (upper) and divergence (lower). Second column: current predictions. Third column: divergence estimates. Fourth column: posterior divergence z-values.
  • Figure 2: First column: ground truth predictions (upper) and divergence (lower). Second column: current predictions. Third column: divergence estimates. Fourth column: posterior divergence z-values.
  • Figure 3: First column: current predictions. Second column: divergence estimates. Third column: posterior divergence z-values.
  • Figure 4: Helmholtz decomposition of vector field $F$. Left: original vector field $F$. Top-right: divergence component of Helmholtz decomposition of $F$. Bottom-right: vorticity component of Helmholtz decomposition of $F$.

Theorems & Definitions (27)

  • Definition 2.1: Helmholtz decomposition, bhatia2013helmoltzhodge
  • Proposition 3.0
  • Proposition 3.0
  • Proposition 3.1
  • Proposition 4.0
  • Proposition 4.0
  • Proposition 4.0
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 17 more