Temperature-Conditioned Diffusion Model

• Temperature-conditioned diffusion models are generative models that explicitly condition on temperature fields and ancillary geophysical covariates to improve data reconstruction.
• They employ diffusion-based methods, such as sparse interpolation and residual diffusion, to enhance the resolution and physical consistency of temperature maps.
• Evaluations on metrics like RMSE, SSIM, and LPIPS demonstrate these models’ ability to capture fine-grained spatial patterns compared to classical interpolation techniques.

A temperature-conditioned diffusion model is a diffusion-based generative or reconstruction model in which temperature is not merely an output variable but an explicit condition on the reverse process. In current arXiv usage, this usually denotes models that generate or reconstruct air temperature, land surface temperature, or climate fields from temperature observations, monthly means, or low-resolution temperature inputs, often together with auxiliary geophysical covariates such as LST, LULC, DEM, or sparse sensor masks (Dai et al., 2024, Tsao et al., 26 May 2025, Zhang et al., 8 Nov 2025). A distinct but related usage treats “temperature” as an inverse-temperature or sampling-temperature control parameter that modulates diffusion dynamics, resampling, or reward tilting rather than conditioning on a physical temperature field (Takahashi et al., 13 Apr 2026, Su et al., 17 Aug 2025).

1. Definition and conceptual scope

In the meteorological, remote-sensing, and climate-emulation literature, a temperature-conditioned diffusion model is a conditional DDPM- or latent-diffusion-type model that learns a prior over temperature fields and then samples from a posterior induced by partial observations or physically relevant covariates. The conditioning signal may be sparse temperature observations and masks, land surface temperature, low-resolution temperature grids, monthly mean temperature maps, or coarse atmospheric predictors (Tsao et al., 26 May 2025, Dai et al., 2024, Bassetti et al., 2023).

The conditioning target varies by application. In sparse interpolation, the objective is to reconstruct a full 2D temperature field from partial observations, formalized as sampling from $p(x \mid y, M)$ where $x$ is the full-state field and $M$ is an observation mask (Tsao et al., 26 May 2025). In urban air-temperature prediction, the target is $2\,\mathrm{m}$ air temperature $T_a$ at $100\,\mathrm{m}$ ground separation distance, generated from LST, LULC-derived features, and metadata (Dai et al., 2024). In land-surface downscaling, the target is high-resolution LST $\mathbf{T}_{HR}$ conditioned on low-resolution LST $\mathbf{T}_{LR}$ and high-resolution geophysical priors $\mathbf{G}_{HR}$ (Zhang et al., 8 Nov 2025). In climate emulation, a spatio-temporal diffusion model is conditioned on monthly averages of temperature or precipitation on a $96 \times 96$ global grid and generates 28 daily fields that are realistic and consistent with those averages (Bassetti et al., 2023).

A plausible implication is that “temperature-conditioned” has become an umbrella term spanning at least three regimes: conditional reconstruction of temperature fields, conditional generation of temperature from related physical observables, and algorithmic control of diffusion dynamics via a temperature-like scalar. Because these regimes use the same diffusion vocabulary but different semantics, the term is inherently polysemous.

2. Probabilistic formulations and conditioning operators

A recurrent formulation is Bayesian posterior sampling. For sparse interpolation over the Southern Great Plains, the problem is posed as reconstructing a full $x$0 temperature field $x$1 from sparse observations $x$2, with conditional inference goal

$x$3

where $x$4 is learned by a DDPM and conditioning is applied through mask-based inpainting and prekriging (Tsao et al., 26 May 2025). The reverse chain uses RePaint-style mask conditioning:

$x$5

with

$x$6

and observed pixels injected at the correct diffusion noise level via

$x$7

The model augments the mask using ordinary kriging: a prekriged field $x$8 is computed, pixels with kriging uncertainty below the $x$9th percentile among unknowns are promoted to “known,” and mask-conditioned DDPM is then run with $M$0 (Tsao et al., 26 May 2025).

A second pattern is physically anchored interpolation between temperature variables. In DiffTemp, the standard terminal pure-noise state is replaced by the LST latent, yielding a deterministic LST-anchored schedule

$M$1

with $M$2 and $M$3 (Dai et al., 2024). The reverse model uses $M$4-prediction, conditioned through ControlNet residuals derived from LST, LULC images, and metadata embeddings. The reverse update is

$M$5

This schedule is explicitly intended to constrain generation to physically plausible $M$6 near LST (Dai et al., 2024).

A third pattern is residual diffusion around a low-resolution temperature baseline. PGDM writes

$M$7

and formulates the task as

$M$8

Its ResShift forward process is centered on the residual $M$9 rather than on pure Gaussian corruption:

$2\,\mathrm{m}$0

with reverse mean parameterized through a denoised estimate

$2\,\mathrm{m}$1

The training loss is direct MSE on $2\,\mathrm{m}$2 rather than $2\,\mathrm{m}$3- or $2\,\mathrm{m}$4-prediction (Zhang et al., 8 Nov 2025).

DiffESM uses a different conditional structure. It is a continuous-time diffusion model with $2\,\mathrm{m}$5-parameterization, conditioned on a spatial map of monthly mean temperature or precipitation, the day of the year that the 28-day sequence begins on, and the diffusion time (Bassetti et al., 2023). The paper does not report an explicit aggregation penalty enforcing exact monthly mean consistency; consistency is learned implicitly from conditioning. This distinguishes it from methods that directly enforce pixelwise or coarse-grid reconstruction constraints.

3. Architectural realizations

The literature contains several distinct architectural realizations, but all retain the basic diffusion decomposition into a learned prior and a conditional reverse operator.

System	Conditioning signal	Backbone
Sparse interpolation	Sparse observations, binary mask, prekriged field	OpenAI guided-diffusion UNet (Tsao et al., 26 May 2025)
DiffTemp	LST, RGB, NDVI, NDBI, NDWI, metadata	Stable Diffusion VAE + U-Net + ControlNet (Dai et al., 2024)
PGDM	LR LST, reflectances, NDVI/NDWI/NDMI, DEM, LULC	Dual-branch UNet-like encoder–decoder (Zhang et al., 8 Nov 2025)
PDE-informed downscaling	ERA5 predictors, static high-resolution maps	Residual latent diffusion model with VAE and UNet-like denoiser (Rosu et al., 27 Oct 2025)
DiffESM	Monthly mean map and day-of-year	Fully convolutional spatio-temporal U-Net (Bassetti et al., 2023)

In sparse interpolation, the DDPM uses OpenAI’s guided-diffusion UNet backbone, predicts $2\,\mathrm{m}$6 rather than $2\,\mathrm{m}$7 or a score, learns $2\,\mathrm{m}$8 with learn_sigma=True, and is trained with $2\,\mathrm{m}$9 diffusion steps, multi-resolution attention at $T_a$0, $T_a$1, $T_a$2, and $T_a$3, Adam with learning rate $T_a$4, batch size $T_a$5, and $T_a$6 epochs for $T_a$7 inputs (Tsao et al., 26 May 2025). Inpainting uses timestep re-spacing to $T_a$8 steps and RePaint resampling with $T_a$9, $100\,\mathrm{m}$0.

DiffTemp is a latent diffusion model built on Stable Diffusion with ControlNet conditioning. The denoising U-Net is pretrained on satellite imagery via DiffusionSat and fine-tuned here; ControlNet ingests conditioning images and injects residuals at multiple scales. The optimizer is Adam with learning rate $100\,\mathrm{m}$1, batch size $100\,\mathrm{m}$2, and training lasts $100\,\mathrm{m}$3 steps total, with fine-tuning $100\,\mathrm{m}$4 steps per city (Dai et al., 2024).

PGDM adopts a dual-branch encoder–decoder. One branch is state-aware and processes $100\,\mathrm{m}$5 through time-adaptive ResBlocks and Multi-Head Non-Local attention; the second branch processes $100\,\mathrm{m}$6. The decoder uses PixelShuffle upsampling and U-Net skip connections, with a final residual head predicting $100\,\mathrm{m}$7 (Zhang et al., 8 Nov 2025). The selected configuration uses base width $100\,\mathrm{m}$8 because it gives RMSE $100\,\mathrm{m}$9 with $\mathbf{T}_{HR}$0 parameters and $\mathbf{T}_{HR}$1 FLOPs, while $\mathbf{T}_{HR}$2 improves RMSE to $\mathbf{T}_{HR}$3 but raises complexity to $\mathbf{T}_{HR}$4 parameters and $\mathbf{T}_{HR}$5 FLOPs (Zhang et al., 8 Nov 2025).

The PDE-informed downscaling model uses a VAE to encode high-resolution temperature, a pre-trained reference UNet upscaler, and a latent diffusion denoiser that learns the residual relative to the upscaler output (Rosu et al., 27 Oct 2025). Because of memory limits, only the final $\mathbf{T}_{HR}$6 million parameters are updated during fine-tuning. DiffESM, by contrast, is fully convolutional in $\mathbf{T}_{HR}$7 format, with interleaved temporal and spatial convolution layers, no self-attention, four downsampling/upsampling levels, and per-level channel widths $\mathbf{T}_{HR}$8, $\mathbf{T}_{HR}$9, $\mathbf{T}_{LR}$0, $\mathbf{T}_{LR}$1 (Bassetti et al., 2023).

4. Physics guidance and physically informed conditioning

The major technical divergence among temperature-conditioned diffusion models concerns whether physics enters only through conditioning variables or through the loss itself.

DiffTemp relies on an LST-anchored latent schedule and on physically interpretable covariates rather than on an explicit physics loss. LST is treated as a “physical boundary,” while RGB, NDVI, NDBI, NDWI, and metadata encode urban fabric, seasonality, and geolocation (Dai et al., 2024). The paper states that no explicit clipping is described; instead, anchoring the terminal state at LST and conditioning on LULC plus metadata act as physical priors. This design also enables counterfactual simulation: edited RGB/LULC can be passed through a separate RGB $\mathbf{T}_{LR}$2 LST diffusion model, and the modified LST and RGB are then fed to DiffTemp to estimate $\mathbf{T}_{LR}$3 under altered urban layouts (Dai et al., 2024).

PGDM is more explicit. It grounds its conditioning variables in the surface energy balance

$\mathbf{T}_{LR}$4

and treats $\mathbf{T}_{LR}$5, $\mathbf{T}_{LR}$6, $\mathbf{T}_{LR}$7, and $\mathbf{T}_{LR}$8 as proxies for albedo, emissivity, vegetation cover, aerodynamic resistance, surface resistance, and related SEB terms (Zhang et al., 8 Nov 2025). Although the analytic SEB residual is not directly evaluated during training, the model monitors physical consistency using an energy-conservation degradation operator $\mathbf{T}_{LR}$9 and

$\mathbf{G}_{HR}$0

The paper emphasizes that this term is used in evaluation rather than training (Zhang et al., 8 Nov 2025).

The PDE-informed latent diffusion model goes further by incorporating a physics-informed loss directly into training:

$\mathbf{G}_{HR}$1

Its physical constraint is not a full prognostic PDE but a diagnostic flux-ratio consistency between coarse and generated fine fields, derived from an effective advection–diffusion balance and computed in decoded pixel space (Rosu et al., 27 Oct 2025). The reported effect is specific: conventional residual LDM training already yields small PDE residuals, while the added loss further reduces the flux-ratio discrepancy and improves spectral consistency, though $\mathbf{G}_{HR}$2 “typically has slightly higher RMSE and lower $\mathbf{G}_{HR}$3 than the best purely statistical baselines” (Rosu et al., 27 Oct 2025).

DiffESM occupies the opposite end of the spectrum. It conditions on monthly mean maps and day-of-year labels but does not introduce an explicit consistency loss

$\mathbf{G}_{HR}$4

which the paper presents only as a reference form not used in training (Bassetti et al., 2023). This suggests a broader methodological split: some temperature-conditioned diffusion models encode physics via priors and covariates, whereas others regularize the reverse process or decoded field by explicit diagnostic constraints.

5. Applications, empirical behavior, and uncertainty

The application space spans sparse-data assimilation, urban microclimate mapping, LST downscaling, atmospheric downscaling, and climate emulation.

For sparse interpolation, four methods are compared: base diffusion, KrigSCD, inverse distance weighting, and conditional Gaussian simulations. Evaluation covers $\mathbf{G}_{HR}$5, $\mathbf{G}_{HR}$6, $\mathbf{G}_{HR}$7, $\mathbf{G}_{HR}$8, and $\mathbf{G}_{HR}$9 known coverage (Tsao et al., 26 May 2025). KrigSCD achieves the lowest LPIPS across all coverages, with average LPIPS $96 \times 96$0 versus $96 \times 96$1 for base diffusion, $96 \times 96$2 for IDW, and $96 \times 96$3 for CGS. By $96 \times 96$4 known coverage, reconstructions are visually close to ground truth, with LPIPS $96 \times 96$5 for KrigSCD. Pixelwise errors behave differently: at $96 \times 96$6 known, RMSE is $96 \times 96$7 for IDW, $96 \times 96$8 for CGS, $96 \times 96$9 for base diffusion, and $x$00 for KrigSCD, while by $x$01 known KrigSCD reaches RMSE $x$02 (Tsao et al., 26 May 2025). The paper’s interpretation is explicit: classical methods minimize RMSE/MAE by design at very low coverage, but diffusion models better recover spatial patterns, urban cold spots, gradients, and texture.

DiffTemp reports same-resolution performance on LSTAT-20K with RMSE $x$03, MAE $x$04, and SSIM $x$05, compared with Random Forest at RMSE $x$06, MAE $x$07, SSIM $x$08; Gradient Boosting at RMSE $x$09, MAE $x$10, SSIM $x$11; Linear Regression at RMSE $x$12, MAE $x$13, SSIM $x$14; and MLP at RMSE $x$15, MAE $x$16, SSIM $x$17 (Dai et al., 2024). In super-resolution, the model achieves RMSE $x$18, MAE $x$19, SSIM $x$20 when downsampled $x$21 at $x$22 is used as an extra condition, and Point SR yields RMSE $x$23, MAE $x$24, SSIM $x$25 when $x$26 points from $x$27 emulate station measurements (Dai et al., 2024). The noise-schedule ablation is especially decisive: a pure-noise terminal schedule yields RMSE $x$28, MAE $x$29, SSIM $x$30, whereas the LST-anchored schedule yields RMSE $x$31, MAE $x$32, SSIM $x$33 (Dai et al., 2024).

PGDM reports the strongest benchmark detail. On Landsat_CN20 test data, it attains RMSE $x$34 and SSIM $x$35 at $x$36, and RMSE $x$37 and SSIM $x$38 at $x$39; the corresponding $x$40 values are $x$41 and $x$42 (Zhang et al., 8 Nov 2025). On Landsat_GLB it reaches RMSE $x$43 and SSIM $x$44, and on ASTER_GLB RMSE $x$45 and SSIM $x$46, outperforming bilinear interpolation, kernel-driven methods, DCF, and MoCoLSK-Net across the reported datasets (Zhang et al., 8 Nov 2025). Its stochasticity is also exploited for self-assessment: with $x$47 samples, the scene-level mean diffusion standard deviation $x$48 has a strong positive linear correlation with actual scene-level MAE,

$x$49

The PDE-informed downscaling model evaluates both statistical and physics-aware scores. It reports that $x$50 achieves the best physics-aware scores, specifically the lowest $x$51 and the lowest median and tightest interquartile range in $x$52, while visual comparisons show preserved fine filamentary structures and reduced speckle relative to the base residual LDM (Rosu et al., 27 Oct 2025). DiffESM, finally, is assessed through spatial maps and histograms of Monthly Hot Streak, Monthly Hot Days, 90th Quantile Values, Monthly Dry Spell, Monthly Dry Days, and SDII; generated differences resemble validation-test differences, indicating that the emulator reproduces the ESM’s spatio-temporal distributional characteristics, with closer agreement for temperature than for precipitation and slight under-prediction bias in precipitation (Bassetti et al., 2023).

6. Ambiguities, limitations, and related meanings of “temperature”

A common misconception is that “temperature-conditioned diffusion model” always means a generative model conditioned on a temperature field. In several lines of work, “temperature” is instead a control parameter governing sampling sharpness or the effective dynamics of the reverse process.

In the statistical-mechanics analysis of discrete diffusion models, the forward kernel can be written in a Boltzmannized form with effective inverse temperature

$x$53

which controls coupling between $x$54 and $x$55 (Takahashi et al., 13 Apr 2026). The paper identifies a speciation transition at

$x$56

and a collapse transition at the REM threshold $x$57. This temperature is not meteorological; it is an effective inverse temperature induced by the noise schedule.

In inference-time scaling with SMC, temperature again denotes reward sharpness:

$x$58

with adaptive temperature

$x$59

used to down-weight unreliable early rewards and emphasize later, more reliable ones (Su et al., 17 Aug 2025). This is orthogonal to classifier-free guidance and unrelated to physical temperature fields.

There is also an older physical-transport usage in which diffusion itself is conditioned by thermodynamic temperature. Within the Haken–Strobl–Reineker SQLE, the long-time diffusion coefficient obeys

$x$60

for diagonal dynamical disorder, and

$x$61

for diagonal plus off-diagonal disorder under white-noise, classical-bath assumptions (Barford, 2024). Here the model is not a generative diffusion model at all; it is a transport theory in which temperature modulates dephasing rates and therefore the diffusion coefficient.

Across the generative literature, the main limitations are explicit. Sparse interpolation in the SGP domain uses a discretized $x$62–$x$63 pixel range and no explicit physical constraints, which the paper identifies as a limitation for operational meteorology (Tsao et al., 26 May 2025). DiffTemp notes possible limitations during extreme heat waves and unusual wind regimes because wind and humidity are absent from conditioning, and further validation with in-situ data is needed (Dai et al., 2024). The PDE-informed downscaler approximates advection direction from the temperature gradient rather than explicit winds in the loss term and is evaluated only over Italy (Rosu et al., 27 Oct 2025). PGDM omits explicit atmospheric and radiation fields, using DEM and LR LST as proxies, and its evaluation follows an upscaling–downscaling protocol (Zhang et al., 8 Nov 2025). DiffESM does not enforce exact monthly-mean consistency, models temperature and precipitation separately, and reports only one ESM family and a limited scenario set (Bassetti et al., 2023).

Taken together, these works indicate that the phrase “temperature-conditioned diffusion model” should be interpreted contextually. In contemporary geoscientific machine learning it usually denotes a conditional generative model for temperature fields, often strengthened by geophysical priors or physics-aware losses. In discrete diffusion theory and inference-time scaling, it instead refers to a temperature-like scalar that governs diversity, fidelity, or phase transitions in the generative process itself.