Dandelion: A Spherical Flower for Neural Simulation of Planetary Dynamics

Till Muser; Giovanni Abati; Ivan Dokmanić

cs.LGAug 27, 2026

Dandelion: A Spherical Flower for Neural Simulation of Planetary Dynamics

Authors: Till Muser, Giovanni Abati, Ivan Dokmanić

Organizations: University of Basel

Abstract

Many dynamical processes unfold on the sphere but the default scientific machine learning architectures are Euclidean. Applying these architectures on a regular lat-lon grid causes problems: Cartesian convolutions become distorted at high latitude; 2D FFTs in Fourier neural operators incorrectly assume double periodicity; Cartesian positional encodings in ViTs distort spherical geodesic distances. Recent work moves towards natively spherical primitives, including spherical convolutions (e.g., DeepSphere or DISCO), Spherical Fourier Neural Operators (SFNOs), and geodesic attention. Here we propose Dandelion, a spherical version of Flower, a warp-based neural PDE solver. Layers of Dandelion predict a tangent-plane displacement and transport features along great circles. We obtain a U-Net-like structure by implementing hierarchical pooling entirely in the spherical-harmonic domain. There are thus no convolutions: spatial mixing is achieved only through spherical coordinate changes, or warps. To compare Dandelion with existing spherical architectures, we release an evolving benchmark suite of challenging, natively-spherical PDE datasets including a modified Galewsky jet, anomalous chained turbulence, Cahn-Hilliard decomposition, spherical Riemann shocks, Held-Suarez dry atmospheric transport and global ocean dynamics. This new benchmark fills the gap in existing spherical datasets which are either too small and stylized, or much too large (ERA5) for model iteration. Dandelion is best or second-best on every dataset, and the gap to non-warp baselines widens with resolution: at $256\times 512$ , Dandelion and Flower2D occupy the top two slots in both single-step prediction and rollout.

Figures & tables

Figure 1: Representative snapshots from each dataset used in this paper: (a) barotropic-jets (vorticity), (b) active-turbulence (vorticity), (c) cahn-hilliard ( $\phi_{0}$ ), (d) shock-caps (momentum $u$ ), (e) dry-atmosphere-3D (zonal wind $u$ at various pressure levels), (f) global-ocean-3D (temperature $T$ at depth levels), and (g) planetswe (azimuthal velocity). Datasets (a) – (f) are newly introduced with this paper, while (g) is taken from the The Well benchmark suite [ 37 ] .

Panel of Figure 2 — Figure 2: Two schemes for defining a sampling displacement on the sphere $S^{2}$ . ( 2(a) ) Lat-lon displacement: an angular offset $(d\phi,d\theta)$ (light blue arrow) is added to the chart coordinates of the query point $\mathbf{p}$ to obtain the new sampling location $\mathbf{q}$ . ( 2(b) ) Tangent warp: a displacement $(du,dv)$ (green arrow) is predicted in the local tangent plane $T_{\mathbf{p}}S^{2}$ relative to the orthonormal basis $(\mathbf{e}_{N},\mathbf{e}_{E})$ , then mapped to a point $\mathbf{q}\in S^{2}$ via the exponential map (Rodrigues’ formula), which traces the great circle through $\mathbf{p}$ in the predicted tangent direction (red arrow).

Dataset	Description	Runs	Size (GB)
barotropic-jets	Barotropic instability from shallow water jets	$960$	$210$
active-turbulence	Anomalous chained antiferromagnetic turbulence	$760$	$18$
cahn-hilliard	Spinodal decomposition	$576$	$15$
shock-caps	Multi-shock shallow water Riemann problem	$500$	$63$
dry-atmosphere-3D	3D Held-Suarez global atmospheric simulation	$162$	$667$
global-ocean-3D	3D global oceanic simulation	$243$	$248$

Table 1: Summary of the provided spherical PDEs benchmark datasets. Sizes are the compressed HDF5 releases.

Figure 3: 3-step autoregressive rollout on active-turbulence . Only Dandelion manages to capture an increase in vorticity magnitude, all other models stay close to the input and decay.

Figure 4: Polar stereographic projection of a 32-step autoregressive rollout on shock-caps . Both Dandelion and Flower2D stay stable over the long rollout, while the FNO and SFNO degrade, potentially due to Gibbs ringing at the shock fronts. Although the prediction of Flower2D is structurally accurate, there are weak spurious oscillations in the lower-right area.

	Dataset	FNO	SFNO	LR 2 Trnsf.	LS 2 Trnsf.	Flower2D	Dandelion
Next-step	planetswe	0.0125	0.0084	–	–	0.0008	0.0005
	barotropic-jets	0.0597	0.0193	0.0091	–	0.0044	0.0022
	active-turbulence	0.7764	0.5558	0.1580	0.1134	0.1422	0.1055
	cahn-hilliard	1.2052	0.2933	0.1434	0.0748	0.0426	0.0293
	shock-caps	0.2695	0.0814	0.0209	–	0.0160	0.0082
	dry-atmosphere-3D	0.1704	0.1040	–	–	0.0290	0.0274

Table 2: Area-weighted VRMSE for unconditioned 4

\to

1 next-step prediction and 1:20 rollout. Best in bold, second best colored green. See Equation 27 for a definition of the metric.

Grid	FNO	SFNO	LR 2 Trnsf.	LS 2 Trnsf.	Flower2D	Dandelion
64 $\times$ 128	1.00 $\times$	0.99 $\times$	0.90 $\times$	1.31 $\times$	0.99 $\times$	0.90 $\times$
128 $\times$ 256	1.00 $\times$	1.00 $\times$	5.18 $\times$	33.55 $\times$	1.08 $\times$	2.24 $\times$
256 $\times$ 512	1.00 $\times$	1.01 $\times$	6.46 $\times$	42.98 $\times$	1.14 $\times$	2.59 $\times$

Table 3: Median epoch time during the first phase of training (1-step prediction), reported relative to the FNO baseline (lower is faster).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Fields	Physical parameters
barotropic-jets	Zonal velocity Meridional velocity Surface height Relative vorticity	Maximum jet velocity Jet mid latitude Perturbation amplitude Global average depth
active-turbulence	Vorticity	Normalized radius Normalized bandwidth
cahn-hilliard	Order parameter	Interface width Mean composition IC noise variance Sphere radius
shock-caps	Fluid depth Zon. depth-integrated momentum Mer. depth-integrated momentum	Number of caps Velocity scaling
dry-atmosphere-3D	Zonal velocity Meridional velocity Air temperature Surface pressure	Angular velocity Mer. temperature delta Vertical temperature delta
global-ocean-3D	Potential temperature Salinity Face- $x$ velocity Face- $y$ velocity Sea-surface height	GM/Redi background diffusivity Horizontal viscosity Vertical diffusivity Surface temperature restoring Surface salinity restoring

Appendix

Table 4: Summary of the different datasets’ main physical features: fields refer to the target variables, physical parameters are the variables modified over different runs. For each combination of given physical parameters, multiple runs are performed with different seeds.

Figure 5: Evolution of the vorticity along a barotropic-jets trajectory. Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) are shared.

Dataset	Parameter	Units	Values	Count
barotropic-jets	$u_{\textup{max}}$	$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$	$60,70,80,90,100$	5
	$\phi_{\textup{mid}}$	°N	$30,40,50,60$	4
	$h_{\textup{pert}}$	$\mathrm{m}$	$60,240$	2
	$\overline{h}$	$\mathrm{m}$	$8000,10000,12000,14000$	4
	Seed	–	$0,1,2,3,4,5$	6
active-turbulence	$R/\Lambda$	–	$2,4,7,10$	4

Appendix

Table 5: Grid of varying physical and stochastic parameters across the spherical PDE datasets.

Figure 6: Four different final states of trajectories from active-turbulence , representing the corners of the parameter space. Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) differ by plot, as different phases reach different limits.

Figure 7: Four different final states of trajectories from cahn-hilliard , representing the corners of the parameter space. Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) are shared.

Figure 8: Evolution of the Momentum U-variable of a shock-caps trajectory. Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) are shared. Note that the radial lines emanating from the poles are not a visual artifact or a solver bug. Because the local eastward basis vector abruptly reverses direction across the pole, the projection of the continuous physical momentum vector onto this basis results in a sharp sign-flip.

Figure 9: Evolution of the zonal wind at $1000\text{\,}\mathrm{hPa}$ (the lowest model level) along a dry-atmosphere-3D trajectory. Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) are shared.

Figure 10: Evolution of the Temperature variable of an global-ocean-3D trajectory, selected to show the seasonality by plotting a snapshot every three months (most clearly visible in the temperature of the Mediterranean sea). Top row: states depicted on the sphere. Bottom row: states depicted on a latlon grid. Color scale limits (vmin/vmax) are shared.

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