Weather Prediction with Diffusion Guided by Realistic Forecast Processes

Zhanxiang Hua * 1  Yutong He * 2  Chengqian Ma 3  Alexandra Anderson-Frey 1

††footnotetext: *Equal contribution 1Department of Atmospheric Sciences,
University of Washington, USA 2Machine Learning Department, Carnegie Mellon
University, USA 3Department of Applied Mathematics, University of Washington,
USA. Correspondence to: Zhanxiang Hua<EMAIL_ADDRESS>
Preprint.

###### Abstract

Weather forecasting remains a crucial yet challenging domain, where recently
developed models based on deep learning (DL) have approached the performance
of traditional numerical weather prediction (NWP) models. However, these DL
models, often complex and resource-intensive, face limitations in flexibility
post-training and in incorporating NWP predictions, leading to reliability
concerns due to potential unphysical predictions. In response, we introduce a
novel method that applies diffusion models (DM) for weather forecasting. In
particular, our method can achieve both direct and iterative forecasting with
the same modeling framework. Our model is not only capable of generating
forecasts independently but also uniquely allows for the integration of NWP
predictions, even with varying lead times, during its sampling process. The
flexibility and controllability of our model empowers a more trustworthy DL
system for the general weather community. Additionally, incorporating
persistence and climatology data further enhances our model’s long-term
forecasting stability. Our empirical findings demonstrate the feasibility and
generalizability of this approach, suggesting a promising direction for
future, more sophisticated diffusion models without the need for retraining.

##  1 Introduction

Figure 1: Illustration of the proposed three types of guidance that are
relevant to weather forecasting using diffusion models. Our model is capable
of incorporating all three types of guidance to facilitate more realistic
weather forecasting with diffusion models.

Weather forecasting, a key element of scientific computing, is essential for
predicting future weather, especially extreme conditions like floods,
droughts, and hurricanes. This forecasting is crucial across various sectors,
impacting daily life, agriculture, energy, transportation, and industry Bouwer
(2019). Recent advancements in deep learning (DL) have led to data-driven
models that rival traditional Numerical Weather Prediction (NWP) models Bi et
al. (2023). While data-driven DL models are gaining traction, the complexity
and chaos of atmospheric systems pose significant challenges to these models,
often leading to difficulties in accurately capturing the nonlinear nature of
weather patterns. In addition, most state-of-the-art DL models are exceedingly
complex, making them expensive and difficult for low-resource users to train.
More often, DL models are trained for weather prediction at a specific future
time, requiring the retraining of multiple models if the existing
autoregressive forecast falls short of the desired forecast period. Last but
not least, once the model training is completed, it is extremely difficult to
modify the model parameters or incorporate additional constraints such as
physical guidance from NWP models at inference time, causing unphysical
hallucinations and unreasonable predictions that do not obey natural physical
laws Selz & Craig (2023). As a result, even though these DL models have
achieved superior performance in certain benchmarks Rasp et al. (2023), their
reliability in real-world forecasting workflows remains questionable,
particularly in long-range forecasts Chattopadhyay & Hassanzadeh (2023),
contributing to trust issues in these weather models McGovern et al. (2024).

Diffusion models (DM) have gained popularity in recent years as AI-generated
art platforms like Midjourney and OpenAI’s DALL-E have grown in popularity
(Ramesh et al., 2021). Built with flexible probabilistic inference properties,
DMs not only showcase high quality samples but also demonstrate remarkable
controllability that does not require retraining of the model (Chung et al.,
2023; Yu et al., 2023; He et al., 2023). As the image generation domain
witnesses tremendous development of DM applications, however, there has been
limited research utilizing DMs for weather forecasting. While recent work from
Deepmind Price et al. (2023) showcases DM’s potential in ensemble medium-range
weather forecasting that surpasses traditional methods, little work has been
shown to explore the unique flexibility and controllability of DMs that can
facilitate more accurate and stable predictions in low resource settings.
Moreover, the current literature only views DMs as another black-box proxy for
better prediction, but misses out on the opportunity to establish formal
connections between the probabilistic inference in diffusion models and the
generative modeling view of traditional weather forecasting.

In this paper, we introduce a new weather forecasting model using a
conditional DM, offering the capability to flexibly output weather predictions
for various future time points. The conditional DM forecasting model enables
both iterative and direct forecasting methods within a single framework, where
iterative forecasts build upon each step’s previous prediction, and direct
forecasts generate specific future predictions from the initial input. In
addition, we design the conditional diffusion model that explicitly takes
previous weather observations, non-weather constant variables and the target
lead time as inputs, and treat the target lead time as “class labels”. By
incorporating classifier-free guidance (Ho & Salimans, 2022), our model is
able to gain benefit from learning both the prior and the posterior
distribution of a weather state given the lead time.

To tackle hallucination issues and the long-term stability of data-driven
models, our diffusion framework can also combine various forecasting processes
without the need for retraining. More specificalluy, we enhance this
conditional DM forecasting model by integrating SDEdit Meng et al. (2021), a
tool that allows adjustable noise levels and input guidances, tailored to the
user’s trust in the DM and the guidance. We showcase the versatility of this
feature in three applications: NWP guidance, persistence guidance, and
climatology guidance, each providing unique solutions. This feature is
particularly useful in addressing the limitations of data-driven DL models,
especially in generating long-term and realistic weather patterns. The
flexibility provided by our framework also allows users to choose the most
suitable forecasting method for their needs and establishes a unique balance
between trust and realism, distinguishing it from other applications of DL
models. Furthermore, the framework opens up the potential integration of long-
range forecast model outputs as guidance coupling with DMs primarily designed
for short-term forecasting. This integration aims to produce forecasts with
higher temporal resolution while ensuring a seamless transition in weather
states, eliminating the discontinuity typically encountered when
reinitializing short-term forecast models.

##  2 Related Works

In the domain of weather forecasting, there has been a paradigm shift from
traditional numerical models, which used physical computer models to
numerically solve atmospheric and ocean equations (as summarized by Bauer et
al. (2015)), to deep learning. Recent years have seen the rise of DL in
weather forecasting, with DL-based models now surpassing traditional NWP
models, a trend highlighted by Rasp et al. (2020) and Rasp et al. (2023).
Notable advancements include DL models like KeislerNet (Keisler, 2022)
challenging state-of-the-art NWP models such as the European Centre for
Medium-range Weather Forecasts’s (ECMWF) high-resolution forecast (HRES).
Others, like FourCastNet (Pathak et al., 2022), Pangu-Weather (Bi et al.,
2023), and a series of new models including GraphCast (Lam et al., 2023), FuXi
(Chen et al., 2023b), FengWu (Chen et al., 2023a), and Stormer (Nguyen et al.,
2023), have shown competitive or superior performance, indicating a
significant advancement in the field.

Recent advancements in DL models have been geared towards probabilistic
forecasting to better handle the unpredictability of weather and the butterfly
effect (Lorenz & Haman 1996), leading to a shift from deterministic to
stochastic ensemble forecast systems. These systems, discussed in studies like
Palmer 2019, aim to represent a broader potential variation of weather
forecasts but struggle with extremely high computational costs. To overcome
these issues, diffusion models have been increasingly utilized in
probabilistic forecasting, showing effectiveness in precipitation nowcasting
(Asperti et al. 2023, Gao et al. 2023) and turbulent flow simulation (Kohl et
al. 2023, Cachay et al. 2023). A significant advancement in this field is the
introduction of GenCast (Price et al., 2023), a diffusion-based ensemble model
that has outperformed ECMWF’s ensemble forecasting in most evaluation metrics.

However, the application of Diffusion Models (DMs) in weather forecasting is
underexplored. For example, techniques like SDEdit introduce guidance and
controllability that balances the preservation of input structure guidance
with the production of realistic generated outputs. In this paper, we use this
approach to enhance DMs in weather forecasting without retraining, overcoming
traditional DL models’ limitations by providing realistic long-term forecasts
and outputs adjustable to user trust levels, and thereby advancing the
stability and reliability of weather prediction models.

##  3 Method

###  3.1 Problem Statement

The goal of this paper is to forecast future weather based on observed
weather. The weather, or more formally weather state,
$x\in\mathcal{X}\subset\mathbb{R}^{H\times W\times C_{1}}$ denotes a
collection of variables that describe the atmosphere conditions at different
pressure levels for a given time such as temperature, humidity, pressure field
height, wind speed, etc. Each weather state is represented by a two-
dimensional grid of $H$ latitude and $W$ longitude grid points with $C_{1}$
channels that represent different variables at different pressure levels.
Setting the current time to be $0$, to represent how far into the future this
prediction will extend, we denote the difference between the target future
time and current time as lead time $K\in\mathbb{R}^{+}$. The target future
weather state $x$ with a lead time $K$ is denoted as $x^{(K)}\in\mathcal{X}$.

We assume there exist $C_{0}$ non-weather constant conditions such as terrain
height and land-sea mask for all locations on the $H\times W$ grid and $L$
observed current or prior weather states available. We then represent the non-
weather constant conditions as $y\in\mathcal{Y}\subset\mathbb{R}^{H\times
W\times C_{0}}$, and the $L$ available prior weather states as a spatial-
temporal sequence
$\vec{x}_{\text{ob}}=\left[x^{(k_{1})},x^{(k_{2})},\cdots,x^{(k_{L})}\right]\in\mathcal{X}^{L}\subset\mathbb{R}^{L\times
H\times W\times C_{1}},k_{l}\in\mathbb{R}$.

With the context provided, we define our weather forecasting task as the
following: Given a sequence of observed weather states $\vec{x}_{\text{ob}}$
and constant variables $y$, what are the future weather states $x^{(K)}$ at
lead time $K$? In terms of probabilistic modeling, we would like to obtain

$x^{(K)*}=\operatorname*{arg\,max}_{x^{(K)}}P(x^{(K)}|y,x^{(k_{1})},x^{(k_{2})},\cdots,x^{(k_{L})})$
(1)

###  3.2 Unified Framework

To answer this question, there exist two major forecasting frameworks –
“direct” and “iterative”. The direct forecast aims for a mapping
$f:\mathcal{Y}\times\mathcal{X}^{L}\to\mathcal{X}$ that directly outputs the
predicted future weather state conditioned on all prior observations and
conditions in one step following Eq (1).

The iterative forecast is similar to traditional NWP modeling: the forecast is
generated recurrently by using the intermediate forecast output as part of the
input conditions for the next step of the forecast until the lead time $K$ is
reached. Specifically, suppose
$\vec{x}_{\text{ob}}=[x^{(K_{0})},x^{(K_{0}+\Delta
k)},...,x^{(K_{0}+(L-1)\Delta k)}]$ where $K_{0}$ is the earliest observed
timestamp and $\Delta k$ is the lead time between each observed timestamp, we
can decompose the direct objective into the product of intermediate steps by
chain rule and perform autoregressive sampling with objective:

$x^{(K_{0}+N\Delta k)*}=\operatorname*{arg\,max}_{x^{(K_{0}+N\Delta
k)}}\prod_{n=L}^{N}P(x^{(K_{0}+n\Delta k)}|y,[x^{(K_{0}+j\Delta k)}]_{j<n})$
(2)

For all $j\geq L$, $x^{(K_{0}+j\Delta k)}$ is calculated as $x^{(K_{0}+j\Delta
k)*}$ using Eq (2). When $N=(K-K_{0})/\Delta k$, we can obtain the target
prediction at lead time $K$.

While direct models offer more advantages in parametrization flexibility and
speed, the iterative models allow forecasting weather states beyond the
predefined lead time during training and thus allow the incorporation of
guidance at any instance of lead time. Hence we ask the question: Is there a
way to unify direct and iterative modeling to obtain the advantages from both
types of models?

To achieve the best of both worlds, we propose a generic modeling framework
that can handle both direct and iterative forecasting. First we observe that
as the time progresses, the dependency between the future weather state and
the past weather state gradually decreases. As a result, we make the modeling
assumption that for given $K_{0}$ and $\Delta k$, there exists a large enough
$L$ such that $x^{(K_{0}+M\Delta k)}\perp\\!\\!\\!\perp
x^{(K_{0})}\,|\,x^{(K_{0}+\Delta k)},\cdots,x^{(K_{0}+L\Delta k)}$ for all
$M\geq L+1$.

With this assumption, $P(x^{(K_{0}+n\Delta k)}|y,[x^{(K_{0}+j\Delta
k)}]_{j<n})\approx P(x^{(K_{0}+n\Delta k)}|[x^{(K_{0}+(n-j)\Delta
k)}]_{j=1}^{L})$. Therefore, we can approximate Eq (2) with:

$x^{(K_{0}+N\Delta k)*}\approx\operatorname*{arg\,max}_{x^{(K_{0}+N\Delta
k)}}\prod_{n=L}^{N}P(x^{(K_{0}+n\Delta k)}|[x^{(K_{0}+(n-j)\Delta
k)}]_{j=1}^{L})$ (3)

where $x^{(K_{0}+n\Delta k)}=f(y,x^{(K_{0}+(n-L)\Delta
k)},\cdots,x^{(K_{0}+(n-1)\Delta k)})$ with direct model $f$ trained with the
objective in Eq (1).

With the above approximation, we can achieve both direct and iterative
forecasting with the same modeling framework.

Figure 2: The overall diffusion model network architecture.

###  3.3 Modeling with Conditional Diffusion

To model this conditional distribution, we design a conditional diffusion
model that explicitly incorporates the non-weather constants, the prior
observations and the target lead time as input conditions along side with the
noisy sample produced from the previous diffusion time step. The overall model
architecture is shown in Figure 2 and the detail settings are in the Section
A.1.

More specifically, we denote the $c$-th channel in observed weather $x^{(k)}$
as $x^{(k)}_{c}$ and the $c$-th channel in the noisy sample at diffusion time
step $t$ as $x^{(K)}_{c,t}$. We first perform a sequential channel-wise
concatenation so that the same channels of all observed weather and the noisy
sample as stacked together. Then we concatenate the non-weather constants on
top of all the weather variables to form a vector
$y\bigoplus(x^{(k_{1})}_{1},\cdots,x^{(k_{L})}_{1},x^{(K)}_{1,t})\bigoplus\cdots\bigoplus(x^{(k_{1})}_{C_{1}},\cdots,x^{(k_{L})}_{C_{1}},x^{(K)}_{C_{1},t})$
$\in\mathbb{R}^{H\times W\times(C_{0}+(L+1)C_{1})}$. This vector together with
the target lead time $K$ and the diffusion time step $t$ forms the inputs of
our diffusion model. To capture the periodicity of weather and to handle
varying target lead time, we also create a Fourier embedding of the input lead
time (Vaswani et al., 2017). Since weather patterns typically closely
correlate with time, we explicitly treat the lead time variable as the “class
label” of our conditional diffusion model.

While this predictability usually holds for predictions in the near future, it
becomes less reliable as the forecast extends further into the future.
Moreover, we can also observe that generally the future weather states are
regarded as potential deviations from the previous observations and the
posterior distribution of a weather state given a lead time is fairly
concentrated around the prior weather distribution. As a result, our model can
also gain benefits from learning the prior distribution of a weather state
without conditioning on the lead time. To incorporate the benefit of learning
from both the unconditional and conditional distribution, we use classifier-
free guidance (Ho & Salimans, 2022) in our model training and sampling. In the
ablation study in Section B.2, we show that the larger the target lead time
is, the smaller the guidance scale is required to produce a relatively
accurate forecast.

Following the configurations described above, we train a diffusion model with
total number of diffusion time steps $T$ with the following objective:

$L(\theta)=\mathbb{E}\left[\omega_{i}\|\epsilon_{\theta}(\sqrt{\alpha_{t}}x^{(K)}+\sqrt{1-\alpha_{t}}\epsilon;y,\vec{x}_{ob},t,K)-\epsilon\|^{2}\right]$
(4)

where $\epsilon\sim N(0,I)$, $\epsilon_{\theta}$ is the diffusion model that
predicts the noise for each time step, $\alpha_{t}$ is the predefined
diffusion noise scale scheduling at diffusion time step $t$. Following Rasp et
al. (2020), we also use $\omega_{i}$ as the latitude weighting factor for the
$i$-th latitude to account for the projection scaling from the latitude-
longitude grid to a 2-D Euclidean grid. For inference, we use DDIM (Song et
al., 2020) sampling and details are provided in Section 4 and the appendix.

###  3.4 Guided Weather Forecasting

Besides directly sampling from the trained diffusion model, one can also
easily incorporate various assumptions, constraints and guidance into the
forecast generation. using SDEdit, a technique developed by Meng et al.
(2021). The key idea of SDEdit is to inject a “preliminary version of the
target output” in the middle of the diffusion process as guidance and continue
removing the remaining noise to achieve an output that is both realistic and
faithful to the input guidance. In the task of weather forecasting, the input
guidance can be existing forecasting results that are easy to compute and
follow certain physical constraints but are not necessarily accurate. SDEdit
initializes the diffusion process from the neighborhood of the input guidance
and gradually fills in the corrupted details of the forecast to be more
precise based on the trained diffusion models.

Figure 3: Generate weather forecasts from guidance of various types with our
framework.

Mathematically speaking, let the existing forecast be $\hat{x}^{(K)}$, then
using SDEdit at diffusion time step $t_{0}$ and $\hat{x}^{(K)}$ as input
guidance is equivalent to imposing the assumption:

$\sqrt{\alpha_{t_{0}}}x^{(K)}+\sqrt{1-\alpha_{t_{0}}}\epsilon\sim\mathcal{N}(\sqrt{\alpha_{t_{0}}}\hat{x}^{(K)},(1-\alpha_{t_{0}})I)$

with $\epsilon\sim N(0,I)$. With different choices of the starting diffusion
time step $t_{0}$, users of our system can decide how much they would like to
“trust” the diffusion model versus the existing forecast: Choosing a smaller
$t_{0}$ suggests that the users believe that an accurate prediction lies in
close proximity to the existing forecast $\hat{x}^{(K)}$, and therefore select
a Gaussian distribution with smaller variance surrounding the scaled
$\hat{x}^{(K)}$. Likewise, a larger $t_{0}$ implies that the users perceive
the existing forecasting methodologies as suboptimal and therefore choose to
explore more by selecting a larger variance and using the diffusion model to
generate more content in the forecast. $t_{0}=0$ is equivalent to directly
predicting with the existing forecast, and $t_{0}=T$ is equivalent to directly
sampling from the diffusion model without any guidance.

With different traditional weather forecasting methods, different
$\hat{x}^{(K)}$ and the resulting outputs can provide unique properties. For
example, this framework uses NWP predictions as a physical guide to ensure
realistic outcomes; with persistence guidance, it manipulates how similar the
next forecast is to the previous forecast. And with climatology guidance, it
allows the generation of forecasts that approach the general physical pattern
of weather over longer periods.

##  4 Experiments

###  4.1 Datasets

We trained the our model on a dataset built from the WeatherBench (Rasp et
al., 2020) which is the ERA5 reanalysis dataset (Hersbach et al., 2020) that
was regridded to 5.625° ($H=32,W=64$) using a bilinear interpolation. We also
subsampled the temporal resolution from 1 hour to 6 hours, corresponding to
00z, 06z, 12z and 18z times each day, where 00z means 00:00UTC in Zulu
convention. To provide a basic demonstration of the task, we simply used
$C_{1}=2$ two-dimensional input-output fields in the model: geopotential
height at 500 hPa (Z500) and temperature at 850 hPa (T850). The Z500 fields
are important in identifying the structure of midlatitude weather systems, and
T850 is important in identifying frontal boundaries. We also input $C_{0}=3$
additional prescribed constant fields from Weatherbench: the land-sea mask,
the topographic height, and the latitude. Our model is trained on the dataset
from 1979 to 2015 for which the input is transformed by a min-max scaler and
scaled to the range of $[-1,1]$. Configurations are tuned based on the 2016
dataset and the results are validated on 2017 and 2018 datasets which are
directly comparable to the baseline from WeatherBench.

###  4.2 Implementation Details

We choose $T=1000$ for all DMs. The $\Delta k$ is equal to 6 hours in all
experiments. $C_{0}=3$ and $C_{1}=2$ for both DM iterative and DM direct as
described in Section 4.1. Further details can be found in Appendix A.

In order to investigate the properties of different tasks, we train two
separate lightweight model for iterative and direct tasks respectively even
though the same framework can be applied to both tasks. For the remaining
parts of this paper, we refer to the model trained for the iterative task as
DM iterative and the one trained for the direct task as DM direct. DM
iterative consists of 4 consecutive observations
$\vec{x}_{\text{ob}}=\left[x^{(0)},x^{(1)},x^{(2)},x^{(3)}\right]$ with output
lead time $K\in\\{1,2,3,4\\}$. We choose to output a single lead time $K=1$
for the DM iterative model that is presented in Section 4.4 and the following
sections. DM direct consists of 4 observations
$\vec{x}_{\text{ob}}=\left[x^{(0)},x^{(1)},x^{(2)},x^{(4)}\right]$ with output
lead time $K\in\\{1,2,4,12,20,28,36\\}$.

###  4.3 Evaluation Metrics

We adopt two key metrics from WeatherBench (Rasp et al., 2020) to measure the
quality of the forecasts: the mean latitude-weighted root mean squared error
(RMSE) for assessing the error magnitudes, and the mean anomaly correlation
coefficient (ACC) for assessing spatial anomaly pattern accuracy. RMSE
measures the forecast precision, with lower values indicating greater
accuracy. A perfect ACC score of 1 denotes absolute correlation with actual
weather events, while 0 implies no predictive power. Unlike RMSE, ACC
evaluates how well the forecast anomalies have represented the observed
anomalies, which is crucial for determining if forecasts capture significant
spatial weather variations or just approximate average weather conditions.

###  4.4 Diffusion Forecast without Guidance

We first compare the performance of our proposed DM model to competing
baselines without any input guidance.

##### Baselines

We choose the same baselines mentioned in WeatherBench, which are briefly
introduced as follows: (1) Persistence Forecast, which uses current conditions
to predict the next day’s weather; (2) Climatological Forecast, comprising
Overall Mean Climatology (averaged over data from 1979 to 2016) and Weekly
Mean Climatology, which is the averaging data for each of the 52 calendar
weeks to account for seasonal variations; (3) Operational NWP Model,
specifically the ECMWF’s Integrated Forecast System, which requires
substantial computational resources, exceeding 10,000 CPU cores; (4) Physical
NWP Models, including the T42 Model with a resolution of 310 km, taking 270
seconds per forecast on 36 CPU cores, and the T63 Model with a resolution of
210 km, requiring 503 seconds per forecast on 36 CPU cores; (5) Linear
Regression, which is used for both iterative 6-hour forecasts and direct 3-day
and 5-day forecasts, trained on two weather states; and (6) a Convolutional
Neural Network (CNN) featuring 5 layers, implemented for both iterative and
direct forecasts, similar to the linear regression approach.

Figure 4: RMSE (a) and ACC (c) of 500 hPa geopotential and RMSE (b) and ACC
(d) of 850 hPa temperature for different baselines at 5.625∘ resolution. Solid
lines are iterative forecasts, while dots represent direct forecasts.

##### Results

In Figure 4a and Figure 4b, both DM direct and DM iterative outperform
persistence approaches and weekly climatology in terms of average global
forecast RMSE in 500-hPa height and 850-hPa temperature fields, surpassing the
latter for up to 7 days (DM direct) and 6 days (DM iterative). Our models also
marginally outperform the T42 model for all forecast lengths, with the DM
direct being slightly more accurate at 7 and 9 days. Similar trends are noted
in T850’s RMSE, with the DM models leading the T42 model by a day. Note that
even though T63 and Operational models are superior to our DM models, our
method still has advantages in terms of computational costs.

The ACC scores for Z500 (Figure 4c) mostly mirror the RMSE rankings. DM models
match the weekly climatology’s ACC score at a forecast limit similar to the
T42, but fall below weekly climatology’s ACC just before 7 days. The DM
iterative performs better than the DM direct in earlier forecasts. For T850
(Figure 4d), our models have a clear advantage over the T42. Overall, our
models’ high ACC scores indicate that they effectively produce realistic
spatial weather patterns with accurate disturbance amplitudes.

###  4.5 Diffusion Guided by Realistic Forecast Processes

Having compared the base performance of our DMs to the baselines, we can now
explore how each guidance further improves the performance of the DM models.
Specifically, we focus on three types of guidance: NWP guidance, persistence
guidance and climatology guidance, and integrate them into the base iterative
6-hourly DM model. In general, combining DMs with different input guidance can
provide various advatanges over the orignal models without any retraining.

####  4.5.1 NWP Guidance

Given an NWP forecast, we want to generate weather forecasts from the
iterative DM that are faithful to the trusted NWP model but also retain the
ability to draw samples from a potentially more accurate distribution.

Figure 5: RMSE (a) and ACC (c) of 500 hPa geopotential and RMSE (b) and ACC
(d) of 850 hPa temperature for incorporating IFS T42 and IFS T63 forecasts as
guidance with different $t_{0}$ The dashed line is the basic 6-hourly DM
iterative model, the magenta dot represents IFS T42 and the blue dot
represents IFS T63.

We start with the T42 guidance (Figure 5) which are lines with color gradients
ranging from magenta to yellow. As shown in Section 4.4, both DM iterative and
T42 have 6-hourly time steps with similar performance in both Z500 and T850.
When we perturb the T42 guidance with relatively low noise (i.e., $t_{0}$ less
than 25) which we trust more towards the output of the T42 model, the forecast
produced by DM after denoising from the perturbed T42 forecast is similar to
the performance with the T42 model. However, if we increase the perturbation
of T42 output by up to 200 steps of Gaussian noise, we achieve overall better
performance compared to both T42 and the basic DM iterative model, especially
in longer lead times. The forecast skill for Z500 improved by over a day of
lead time for day 7. As for T850, the performance of T42 is much worse
compared to Z500, such that incorporating the T850 field generally penalizes
the performance in an earlier lead time. But for longer lead time, with a
perturbation of T42 output by 200 steps of Gaussian noise, the forecast skill
for T850 also improved by over a day of lead time for day 7. An example
forecast of T42, DM iterative with T42 guidance, and the difference of RMSE
between the two predictions is shown in Figure 6.

Figure 6: Example fields for 2017-12-24 12UTC initialization time. The top row
shows the forecast produced by T42. The second row shows the forecast produced
by the iterative DM after denoising from the T42 forecast perturbed by 200
steps of Gaussian noise. The last row show the difference of RMSE between the
T42 guided iterative DM forecast and T42 forecast.

As for T63 guidance with 12-hourly time steps which are lines with color
gradients ranging from blue to cyan (Figure 5), the DM iterative incorporates
the T63 guidance once every two time steps. We find that perturbing the T63
guidance with stronger Gaussian noise overall penalizes the performance of the
model relative to the forecast of T63. This is limited by the performance of
the DM which is not robust enough to accurately predict the diffusion
posterior mean or the noise component at any given step in the sampling
process. Forecast samples with T63 guidance and further discussion about this
limitations are in Appendix D. That being said, this framework is highly
flexible as the user is allowed to adjust the amount of noise that represents
their trust relative to the NWP model even for different time steps.
Furthermore, it demonstrates the potential of blending in guidance with
different time steps with the DM forecasts.

####  4.5.2 Persistence Guidance

We use the DM iterative output for the previous lead time as te persistence
guidance and aim to generate forecasts that resemble the previous outputs. The
key idea is that, with this prior imposed, it is less likely to obtain a
diffusion posterior mean with large deviations during the sampling process.

Figure 7: RMSE (a) and ACC (c) of 500 hPa geopotential and RMSE (b) and ACC
(d) of 850 hPa temperature for incorporating previous forecasts from the DM
iterative model itself as guidance with different $t_{0}$. The dashed line is
the basic 6-hourly DM iterative model, and the yellow dot represents an
optimized combination for the amount of perturbation for each lead time. The
configuration of “Persist $t_{0}=$ multi” is in Section C.1

The persistence guidance generally improves the forecast skill in longer lead
times. Looking at the RMSE and ACC of Z500 (Figure 7 (a) and (c)), DM
iterative performs relatively well for up to 4 days when compared to other
persistence guidance. Nevertheless, incorporating the previous forecasts and
perturbing with relatively strong noise (i.e. 500 and 600 steps) in early lead
times still slightly improves the skill of the forecast. Reducing the
perturbation added to the previous forecast, which lets the DM assume the
forecast varies less than its previous forecast, in a longer lead time beyond
5 days, reduces the model error growth. These characteristics are also found
in the T850 field (Figure 7b,d). However, because T850 has a larger spatial
variation in time, the performance advantages that persistence guidance brings
are not as significant as in the Z500 field, such that showing more trust
toward previous forecast output as guidance more heavily penalizes performance
in earlier lead times compared to the Z500 field. Finally, note that the
yellow dot in Figure 7 is a combination of the perturbation added to the
previous forecast with smaller perturbations in shorter lead times and
gradually larger perturbations in longer lead times, which boosts the forecast
performance at all lead times compared to the basic DM iterative model.
Forecast samples and configurations are presented in Appendix C

####  4.5.3 Climatology Guidance

For the climatology guidance task, we want to generate weather forecasts from
the iterative DM model that resemble the climatology, which is the weather
conditions averaged over a period of time. Imposing climatology guidance
reduces the likelihood of obtaining a biased weather prediction that deviates
from the general climate states such as seasonality, which many DL models
struggle with in free-running experiments.

Figure 8: RMSE (a) and ACC (c) of 500 hPa geopotential and RMSE (b) and ACC
(d) of 850 hPa temperature for incorporating “Persist: multi” from Figure 7
and weekly climatology (W_Climo) to the DM iterative model as guidance with
different $t_{0}$. The dashed line is the basic 6-hourly DM iterative model,
and the “multi” in the legend denotes incorporating different $t_{0}$ for each
lead time with the configuration of “W_Climo 144h $t_{0}=$ multi” shown in
Section C.1

We use the weekly climatology as it captures seasonal variation compared to
overall mean climatology. The weekly climatology generally improves forecasts
of the model near or beyond the lead time of the weekly climatology skill
which such skill was shown in baseline (Section 4.4). Before starting to apply
the weekly climatology guidance (Figure 8), persistence guidance is applied to
all runs based on the optimized combination for the amount of perturbation for
each lead time in Section 4.5.2.

Figure 8 (a) and (c) show the performance of the Z500 field, with examples
that applied weekly climatology guidance. Incorporating weekly climatology
guidance starting from day 8 with $t_{0}=300$ can already achieve an RMSE
$<1000$\mathrm{m}^{2}\,\mathrm{s}^{-2}$$ in Z500 at day 14 with an ACC of
greater than 0.5. Similar performance gains can also be found for the T850
field. For experiments that impose stronger weekly climatology guidance
starting either from day 7 or day 8, a more abrupt adjustment to the results
can be observed in the figure. To account for this abruption, we gradually
decrease $t_{0}$ as the lead time increases, and label those runs as “$t_{0}=$
multi”. An example forecast is presented for “W_Climo 144h $t_{0}=$ multi” in
Section C.3. Note that incorporating strong weekly climatology guidance can
lead to the oversight of important diurnal variations in weather forecasts,
thereby diminishing their overall relevance and accuracy. That being said,
users also have the flexibility to incorporate other types of smoothing
climatology with different kernel sizes to suit their needs.

###  4.6 Ablation Study

In our experiments, we also find that the DM’s performance is sensitive to the
number of DDIM sampling steps and classifier-free guidance weight, which is
rarely documented in the related literature. Our findings indicate that for
the specific model and task of predicting two weather states based on two
observed states, as few as 12 steps of sampling are sufficient to obtain
optimal performance. Additionally, our results suggest that a decrease in the
classifier-free guidance weight is necessary to maintain relatively good
performance. The detailed analysis about these findings are in Section B.1 and
Section B.2. Lastly, Appendix D offers a brief analysis of the challenges
encountered when incorporating the T63 guidance and discusses the limitations
of the SDEdit framework in weather forecasting applications.

##  5 Conclusion

We introduce a novel unified framework that uses conditional diffusion models
for weather forecasting. Our method can generate predictions across various
future time points with great flexibility for both iterative and direct
forecasts. In addition, we enhance our method by integrating SDEdit, which is
adaptable to suit diverse weather forecasting needs by providing different
physical guidance without retraining the models. These guidance include
numerical weather predictions, persistence and climatology, which are methods
that generally are used in the weather forecasting community. This framework
allows users achieve trustworthy and competitive results in a low-resource
setting and is adjustable based on their needs and preferences. In addition,
the persistence and climatology guidance also improve the long-term forecast
stability of our diffusion models. Our framework holds the potential to
integrate a broader spectrum of guidance methods, paving the way for more
trustworthy forecasting solutions with diffusion models.

##  6 Impact Statement

Our work proposes a unified framework using diffusion models for future
weather forecasting across various periods. We further explore the potential
of combining realistic forecast processes such as the traditional numerical
weather forecast (NWP) to guide the weather forecasts generated by the
diffusion models. Specifically, our approach allows users to balance their
trust in NWP and diffusion models in producing forecasts. On the bright side,
incorporating NWP guidance into diffusion model forecasts enables users to
anticipate weather system evolution based on prior experiences with such
guidance. This facilitates more applications of DL-based models and improves
the trustworthiness of these models in weather forecasting. However, it is
crucial to remain cautious about the risks associated with generative AI in
weather forecasting. Despite our method offering controllable forecast
generation to minimize unrealistic weather predictions, there is still a
possibility of generating plausible but incorrect forecasts, a challenge that
even NWP models have been striving to overcome for decades. Continuous
monitoring and cross-validation with other forecast sources are essential to
ensure the reliability of our method.

##  7 Acknowledgement

We would like to acknowledge high-performance computing support from Casper
Computational and Information Systems Laboratory (2019) and Derecho: HPE Cray
EX System (https://doi.org/10.5065/qx9a-pg09) provided by NCAR’s Computational
and Information Systems Laboratory, sponsored by the National Science
Foundation.

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##  Appendix A Experiment Details

###  A.1 Diffusion Model Architecture

Our diffusion model network architecture (Figure 2) follows the settings in Ho
et al. 2020, featuring channel depths of 64, 128, and 256 at each respective
level. To condition on the noise level $t$ and lead time $K$, we follow the
implementation in Ho et al. (2020) which creates sinusoidal timestep
embeddings. The noise level $t$ is scaled within the range [0,1] whereas the
lead time $K$ label adds a positive constant to scale greater than a minimum
of 1. The sinusoidal timestep embeddings of $t$ and $K$ are passed through a
2-layer MLP and summed together. Each of the ResNet blocks in the block
applies a further MLP layer to the summation of the aforementioned timestep
embeddings which are then added to the hidden state of the ResNet block.

###  A.2 Diffusion Hyperparameter Settings

The total number of diffusion steps is identical to DDPM (Ho et al., 2020) in
which $T=1000$. Similarly, the $\beta_{t}$ starts at 0.0001 and ends at 0.02
with the linear scheduling. We train our model using classifier-free guidance
with $35\%$ conditional training and $65\%$ unconditional training.

###  A.3 Further Implementation Details

##### Training

Both DM direct and DM iterative are trained on a single Nvidia A100 with a
batch size of 32. Each model training takes approximately 20 hours until no
noticeable performance improvement based on the RMSE of the sampled forecast
from the validation dataset.

##### Inference

The DM iterative with $K=1$, which is shown in the experiments, requires 56
iterative sampling steps to achieve a 14-day (336h) forecast. We select 12 as
the number of DDIM sampling steps and 1.75 as the classifier-free guidance
weight based on ablation study in Section B.1 and Section B.2. With these
settings, generating a 14-day forecast is accomplished in 1.1 seconds on a
single Nvidia A100. The advantage in speed of using DM direct is much more
significant here since this model can produce a 9-day forecast in a single run
with 4 sampling steps. As for SDEdit with $t_{0}=200$, we used 40 DDIM
sampling steps to generate each guided forecast.

##  Appendix B Ablation Study

###  B.1 The Effect of The Numbers of Sampling Steps

Figure 9: The effect of different number of DDIM sampling steps on RMSE Z500
for DM iterative with different lead times $K$. The classifier-free guidance
weight is fixed as 1 in this experiment.

Figure 10: The effect of different number of DDIM sampling steps on RMSE Z500
for DM direct with different lead times $K$. The classifier-free guidance
weight is fixed as 1 in this experiment.

Our lightweight diffusion model, designed to forecast two weather variables
using two input weather variables, demonstrates that the RMSE of Z500 reaches
its minimum within 12 DDIM sampling steps. This finding holds true for both
the DM iterative (Figure 9) and DM direct (Figure 10) methods, which differ in
prior weather conditions and lead times. Notably, when the lead time $K$ is
less than 4, the RMSE of Z500 is minimized at approximately 10 steps. For the
DM direct method, a greater value of K results in a need for fewer DDIM
sampling steps to minimize the RMSE, with 4 steps being optimal in our current
setup when $K$ is greater than 12.

Typically, it is challenging to achieve high-quality results from high-
resolution natural images with a limited number of sampling steps, due to the
high dimensionality of the data distribution. However, in our experiment, the
model we employ is specifically trained to estimate noise for sypnotic scale
variables like Z500 and T850, which exhibit less spatial variation also with a
relatively low resolution of $32\times 64$ latitude-longitude grids. This
suggests that the dimensionality of our data is considerably lower than that
of more complex natural images, and therefore it requires fewer DDIM steps to
achieve high quality predictions. A recent study by DeepMind (Price et al.,
2023) also demonstrates that their model can accurately sample weather
conditions across more than 80 variables at a spatial resolution of $180\times
360$ latitude-longitude grids with 20 steps using the DPM Solver++ 2S Lu et
al. (2022), which supports our findings.

###  B.2 The Effect of The Classifier-Free Guidance Weights

##### DM Iterative

Figure 11: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM iterative with
$K=1$.

Figure 12: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM iterative with
$K=2$.

Figure 13: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM iterative with
$K=3$.

Figure 14: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM iterative with
$K=4$.

Our analysis demonstrates the impact of varying the classifier-free guidance
weight on the Z500 RMSE for the DM iterative model across different numbers of
DDIM sampling steps, as illustrated in Figures 11 to 14. We observe that for
the DM iterative model, a classifier-free guidance weight above 1 and below 2
consistently yields a lower RMSE. Furthermore, it appears that as lead time
$K$ increases, a lower classifier-free guidance weight is needed to attain a
similarly low RMSE. This trend shows the necessity of choosing the classifier-
free guidance weight with respect to the lead time $K$ to optimize the model’s
performance.

##### DM Direct

Figure 15: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with $K=1$.

Figure 16: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with $K=2$.

Figure 17: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with $K=4$.

Figure 18: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with
$K=12$.

Figure 19: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with
$K=20$.

Figure 20: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with
$K=28$.

Figure 21: The effect of different classifier-free guidance weights and
different number of DDIM sampling steps on RMSE Z500 for DM direct with
$K=36$.

We also demonstrate the impact of varying the classifier-free guidance weight
on the Z500 RMSE for the DM direct model across different numbers of DDIM
sampling steps, as illustrated in Figures 15 to 21. We find that similar to
the DM iterative model, as the lead time $K$ increases, a lower classifier-
free guidance weight is needed to attain a similarly low RMSE. For long-range
lead times that have $K$ beyond 20, a classifier-free guidance weight of
around 0.5 is able to attain a lower RMSE for that specific long-range lead
time $K$. This trend potentially suggests that sampling weather states for
longer lead times, which inherently possess greater uncertainty, tends to
align more closely with unconditional sampling, where the lead time $K$ is not
specified.

##  Appendix C Examples of Persistence and Weekly Climatology Guidance
Diffusion

###  C.1 $t_{0}$ Configurations

We provide detailed specification of $t_{0}$ for “persist multi” in Figure 7
and “Climo 144h refined” in Figure 8 as the following:

##### “Persist $t_{0}$= multi”

begins with $t_{0}=700$ for lead times less than 24 hours. For each subsequent
24-hour period, $t_{0}$ is sequentially reduced: It first decreases by 100 for
the 24 hours in the immediate future, and then it further decreases by 200 for
each of the following 24-hour interval. This reduction continues until $t_{0}$
reaches 100 and remains at 100 until the lead time reaches 168 hours. Beyond
this 168-hour lead time, $t_{0}$ is set to 50 for all remaining iterations.

##### “W_Climo 144h $t_{0}=$ multi”

initially employs persistence guidance with the same $t_{0}$ schedule as
outlined in “Persist $t_{0}$= multi”. Starting from the 144-hour lead time, it
transitions to weekly climatology guidance, where $t_{0}$ starts at 250 and is
gradually reduced to 100.

###  C.2 Examples of Persistence Guidance

We present examples of forecasts generated with persistence guidance in Figure
22 and Figure 23.

Figure 22: Similar to Figure 6, but for persistence guidance represented by
“Persist $t_{0}$= multi” in Figure 7 for lead time from 24h to 120h. The top
row represents the baseline DM iterative model for comparison.

Figure 23: Similar to Figure 22 but for lead time from 168h to 264h.

###  C.3 Examples of Climatology Guidance

We present examples of forecasts generated with climatology guidance in Figure
24 and Figure 25.

Figure 24: Similar to Figure 6, but for weekly climatology guidance
represented by “W_Climo 144h $t_{0}=$ multi” in Figure 8 for lead time from
24h to 120h. The top row represents the guidance used which is the weekly
climatology forecast.

Figure 25: Similar to Figure 24 but for lead time from 168h to 264h.

##  Appendix D Failure Case Analysis and Limitations

In Figure 5, we notice that using DM with a very strong NWP model as guidance
may not improve the performance upon the NWP model itself. Examples using T63
as guidance similar to Figure 6 are provided for analysis in Figure 26-29. We
find that those guided predictions generated atmospheric fields, especially
for T850 field, are less smooth in contrast to the more natural field in the
top row of the images produced by T63 forecast.

Furthermore, unlike the application of image synthesis, besides the model’s
performance, there are other limitations or precautions when applying guidance
specifically to weather forecasting. For example, the performance of the
forecast for different weather fields could vary even using the same model.
Thus when applying guidance towards output, $t_{0}$ should be selected
carefully while considering the performance of all fields that the model
produces. Moreover, climatology guidance tends to focus on improving the long-
range forecast stability of the diffusion model. However, there are also other
ways to calculate climatology with different temporal window sizes. Those
different climatologies could require more testing when coupled with different
$t_{0}$. In addition, persistence guidance is also field sensitive. One can
imagine that if a field varies dramatically from one time to the next,
persistence guidance when applied with small $t_{0}$ could dramatically reduce
the forecast skill. We hypothesize that DM forecasting models that are
designed to forecast large-scale weather conditions with smaller $\Delta K$
are more beneficial to apply persistence guidance and leave this analysis to
future works.

Figure 26: Similar to Figure 6, but to demonstrate the limited performance
when using IFS T63 guidance with initialize time of 2017-01-22 12UTC. We
provide detailed failure case analysis in Appendix D.

Figure 27: Similar to Figure 26 but for lead time from 120h to 168h.

Figure 28: Similar to Figure 6, but to demonstrate the limited performance
when using IFS T63 guidance with initialize time of 2017-08-06 12UTC. We
provide detailed failure case analysis in Appendix D.

Figure 29: Similar to Figure 28 but for lead time from 120h to 168h.