Monthly Diffusion v0.9: A Latent Diffusion Model for the First AI-MIP

Kyle J. C. Hall; Maria J. Molina

cs.LGApr 15, 2026

Monthly Diffusion v0.9: A Latent Diffusion Model for the First AI-MIP

Authors: Kyle J. C. Hall, Maria J. Molina

Organizations: Department of Atmospheric & Oceanic Sciences, University of Maryland, College Park, MD 20740

Abstract

Here, we describe Monthly Diffusion at 1.5-degree grid spacing (MD-1.5 version 0.9), a climate emulator that leverages a spherical Fourier neural operator (SFNO)-inspired Conditional Variational Auto-Encoder (CVAE) architecture to model the evolution of low-frequency internal atmospheric variability using latent diffusion. MDv0.9 was designed to forward-step at monthly mean timesteps in a data-sparse regime, using modest computational requirements. This work describes the motivation behind the architecture design, the MDv0.9 training procedure, and initial results.

Figures & tables

Short Name	Name	Level(s)
SKT	skin temperature	surface/single-level
MTPR	mean total precipitation rate	surface/single-level
2D	2 meter dewpoint temperature	surface/single-level
2T	2 meter temperature	surface/single-level
SP	surface pressure	surface/single-level
10U	10 meter U wind component	surface/single-level

Table 1: Variables used for MDv0.9.

Quantity	Description
$x_{t}$	prognostic atmospheric state, $x_{t}\in\mathbb{R}^{42\times 121\times 240}$
$f_{t}$	external forcing fields (SST, SIC, LSM), $f_{t}\in\mathbb{R}^{3\times 121\times 240}$
$s$	static invariant fields (ANOR, ISOR, SLOR, SDOR), $s\in\mathbb{R}^{4\times 121\times 240}$
$m_{t}$	$m_{t}\in[1..12]$ corresponding to months of the year.
$e_{t}^{\mathrm{season}}$	Contractions of sample-wise seasonal coefficients with learned seasonal basis, $SE(m)\otimes SB=e_{t}^{\mathrm{season}}\in\mathbb{R}^{3\times 121\times 240}$
$c_{t}$	time-varying conditioning tensor, $c_{t}=[f_{t};e_{t}^{\mathrm{season}}]\in\mathbb{R}^{6\times 121\times 240}$

Table 2: State (initial and prognostic)

x_{t}

, conditioning

c_{t}

, and latent representation.

Figure 1: Schematic of MDv0.9. Data are denoted by circles, neural networks are denoted by polygons, and subordinate operations are denoted by round-cornered rectangles. Arrows entering laterally indicate data streams used as direct inputs to the indicated operation, and arrows entering from below indicate data streams used as conditioning tensors.

Figure 2: Schematic of spatial conditioning RMS normalization layer.

Figure 3: Schematic of seasonality embedding.

Figure 4: Schematic of an S2-convolution layer. A spectral operator is applied to the internal feature tensor after a spherical harmonic transform, after which the result is inverted onto a new spatial grid.

Figure 5: Schematic of a tensor-product spectral operator. The shapes of the input tensor and operator parameters are denoted below the corresponding objects, and the dimensionality of the intermediate results is displayed above each operation. Here

B

is the minibatch dimension,

C_{\mathrm{in}}

corresponds to the input channels,

L

and

M

to the input zonal and meridional wave modes respectively,

R

to the operator rank, and

C_{\mathrm{out}}

to the output channels.

\odot

corresponds to matrix multiplication (inner product) and

\otimes

to the tensor product (outer product).

Figure 6: Schematic of the encoder. In Fig. 1 , the resampling block in the conditioning pathway is placed outside of the encoder for visual clarity. The residual pathways are omitted from the S2-convolution layers located inside the SpatialConditionalRMSNorm blocks. The decoder uses this same architecture, but with inverted dimensions where appropriate. The predictor uses the same architecture, but with no static fields, and an additional input for DDPM-style noise input.

Component	Description
Encoder $E_{\phi}$	maps $(x_{t},c_{t},s)$ to posterior parameters $(\mu_{t},\log\sigma_{t}^{2})$ where $q_{\phi}(z_{t}\mid x_{t},c_{t},s)=\mathcal{N}(\mu_{t},\mathrm{diag}(\sigma_{t}^{2}))$
Decoder $D_{\theta}$	maps latent state and conditioning back to physical space, modeling $p_{\theta}(x_{t}\mid z_{t},c_{t})$
Seasonal Embedding $SE(m)$	MLP taking inputs $[\sin(2\pi\,m/12),\cos(2\pi\,m/12)]$ , and producing coefficients for a learned latent seasonal basis tensor $SB$ with shapes: input $\ (B,2)\rightarrow$ hidden $(B,64)\rightarrow$ output $(B,3)$
Learned Seasonal Basis $SB$	tensor of shape $(3,3,121,240)$ . Seasonal embedding derived by the tensor product $SE(m)\otimes SB$ , contracting the basis dimension.
Shared spectral operator family	low-rank tensor-product spectral operator with channel mixing $W_{\mathrm{in}}$ , $W_{\mathrm{out}}$ and spectral factors $A,B$
Encoder spectral rank $r_{E}$	64

Table 3: Architecture summary for MDv0.9 at

1.5^{\circ}

resolution: encoder, decoder, and spectral building blocks.

Component	Description
Predictor $v_{\psi}$	conditional latent diffusion model for the next-month latent target
Training target	next-month posterior mean $\mu_{t+1}$ , conditioned on stochastic $z_{t}$
Diffusion steps	$T=15$
Noise schedule	cosine $\beta$ -schedule with offset $s=0.008$
Time encoding	sinusoidal diffusion-time embedding followed by MLP
Time-embedding width	32-dimensional sinusoidal embedding projected to 5 spatial channels

Table 4: Architecture summary for MDv0.9 at

1.5^{\circ}

resolution: latent diffusion predictor and conditioning pathway.

\tilde{\beta}_{k}=\frac{1-\bar{\alpha}_{k-1}}{1-\bar{\alpha}_{k}}\,\beta_{k}

Algorithm 1 Conditional latent DDPM with

v

-prediction. Physical time

t

indexes the monthly evolution, while diffusion time

k

indexes the internal denoising chain used to sample the conditional transition.

Figure 7: Cosine-latitude-weighted annual-mean skin temperature from each of the four sea-surface temperature forced ensemble experiments.

Figure 8: Annual-mean climatology of the MDv0.9 historical ensemble mean, differences between ERA5 and MDv0.9 historical ensemble mean, and differences between MDv0.9 historical +2K and MDv0.9 historical ensemble means.

Figure 9: ERA5 vs MDv0.9-historical regression on forced Niño-3.4

Figure 10: Regression onto the leading EOF of Z500 over 0–90°N for ERA5 and the climatology-forced MDv0.9 run.

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