# Fourier Neural Operators for Arbitrary Resolution
Climate Data Downscaling

Qidong Yang<EMAIL_ADDRESS>
Mila Quebec AI Institute, Montreal, Canada
New York University, New York, USA Alex Hernandez-Garcia
Mila Quebec AI Institute, Montreal, Canada
University of Montreal, Montreal, Canada Paula Harder
Fraunhofer ITWM, Kaiserslautern, Germany
Mila Quebec AI Institute, Montreal, Canada Venkatesh Ramesh
Mila Quebec AI Institute, Montreal, Canada
University of Montreal, Montreal, Canada Prasanna Sattegeri
IBM Research, New York, USA Daniela Szwarcman
IBM Research, Brazil Campbell D. Watson
IBM Research, New York, USA David Rolnick
Mila Quebec AI Institute, Montreal, Canada
McGill University, Montreal, Canada

###### Abstract

Climate simulations are essential in guiding our understanding of climate
change and responding to its effects. However, it is computationally expensive
to resolve complex climate processes at high spatial resolution. As one way to
speed up climate simulations, neural networks have been used to downscale
climate variables from fast-running low-resolution simulations, but high-
resolution training data are often unobtainable or scarce, greatly limiting
accuracy. In this work, we propose a downscaling method based on the Fourier
neural operator. It trains with data of a small upsampling factor and then can
zero-shot downscale its input to arbitrary unseen high resolution. Evaluated
both on ERA5 climate model data and on the Navier-Stokes equation solution
data, our downscaling model significantly outperforms state-of-the-art
convolutional and generative adversarial downscaling models, both in standard
single-resolution downscaling and in zero-shot generalization to higher
upsampling factors. Furthermore, we show that our method also outperforms
state-of-the-art data-driven partial differential equation solvers on Navier-
Stokes equations. Overall, our work bridges the gap between simulation of a
physical process and interpolation of low-resolution output, showing that it
is possible to combine both approaches and significantly improve upon each
other.

Keywords: climate science, climate modeling, super-resolution, downscaling,
neural operator

##  1 Introduction

Climate simulations are running hundreds of years ahead to help us understand
how climate changes in the future. Complex physical processes inside climate
dynamical systems are captured by partial differential equations (PDEs), which
are extremely expensive to solve numerically. As a result, running a long-term
high-resolution climate simulation is still not feasible within the
foreseeable future (Balaji, 2021), even with the current fast-increasing
computational power. Given neural networks’ fast forward inference speed, deep
learning has been applied to speed up climate simulations in the following two
directions.

First, neural networks are used as surrogate solvers to circumvent expensive
numerical methods. More specifically, neural networks are trained with climate
simulation data to approximate complex climate systems serving as climate
model emulators. In recent years, neural network emulators have been
successfully developed for modeling cloud, aerosol, and water systems (Beucler
et al., 2019; Harder et al., 2022a; Tran et al., 2021). Second, deep learning
is also used to predict high-resolution versions of the lower-resolution
outputs produced by climate simulators. Such a process is known as downscaling
in the climate science community and it resembles the problem of image super-
resolution in the machine learning community. The recent works by Höhlein et
al. (2020); Price and Rasp (2022); Groenke et al. (2020) show that deep
learning has achieved excellent performance at climate data downscaling on
variables such as near-surface wind fields, precipitation, and temperature.

Limited by classic neural networks, which map between finite-dimensional
spaces, neural network downscaling models typically have fixed input and
output sizes. For a single trained model, it can only downscale input samples
with a pre-defined upsampling factor. Inspired by the recent success of
Fourier neural operator (Li et al., 2021, FNO) for solving PDEs regardless of
resolution, here we propose a novel FNO based zero-shot climate simulation
data downscaling model, which is able to downscale input samples to arbitrary
unseen high resolution by training only once on data of a low upsampling
factor.

We evaluate our FNO downscaling model in three experiments: PDE integration,
PDE solution downscaling and observational climate quantity downscaling. The
PDE involved in the first two experiments is the Navier-Stokes equations, the
central equation in most climate simulators, which describes physics status of
a moving fluid (e.g., ocean or atmosphere). The observational climate quantity
used in this work is the total column water content which we derived from the
climate reanalysis data base ERA5 (Hersbach et al., 2020). Climate downscaling
models are generally applied to PDE based climate simulation as a post-
processing tool to cheaply generate high-resolution simulation from a fast-
running low-resolution numerical climate simulation model. Our FNO downscaling
model fits this application well since smooth simulation data have a succinct
representation in the Fourier basis, making it easier to be modeled by FNO
with a truncated Fourier series. Evaluation on ERA5 water content data intends
to examine to what extent our model can capture less smooth and noisy
observational data.

Downscaling experiments on Navier-Stokes solution data and water content data
show that our model achieves great performance not only on the learned
downscaling (i.e., the upsampling factor the model is trained on) but also on
zero-shot downscaling (i.e., even higher upsampling factor unseen during
training). The performance is even further improved when a softmax constraint
layer (Harder et al., 2022b) is stacked at the end of our model architecture
to enforce conservation laws. In the PDE integration experiment, our model is
used to downscale low-resolution solution from a numerical Navier-Stokes
equation solver. The downscaled solution obtains significantly higher accuracy
than that from an FNO equation solver—one of the state-of-the-art data-driven
solvers (Li et al., 2021). These results validate our model’s potential to
cheaply and accurately generate arbitrarily high-resolution climate simulation
with fast-running low-resolution simulation as input.

#### Contributions

Our main contributions can be summarized as follows:

  * •

To our best knowledge, we are the first to use FNOs for climate downscaling
and to design an arbitrary-resolution downscaling model.

  * •

Our FNO downscaling model performs significantly better than state-of-the-art
deep learning-based downscaling models.

  * •

When trained on lower-resolution data and tested zero-shot on higher-
resolution data, our method outperforms prior methods trained directly on
higher-resolution data.

  * •

Combining our FNO downscaling model with a low-resolution physical solver, the
resultant high-resolution solution outperforms that from a state-of-the-art
data-driven solver.

##  2 Related Work

###  2.1 Physics-Constrained Deep Learning for Climate System Emulation

Due to their high approximation capacity and fast inference speed, neural
networks have been widely applied for climate system emulation (McCoy et al.,
2020; Watson-Parris, 2021; Kasim et al., 2021). In such settings, it is
essential for the output of a neural network not merely to be close to the
ground truth, but also consistent with certain physical laws, which is
important both for many downstream applications and for trustworthiness.
Various works have attempted to embed physics constraints into neural network
emulators by either adding violation penalty terms to the loss function (i.e.,
soft-constrained) or carefully designing a physics-preserving model structure
(i.e., hard-constrained). Beucler et al. (2021) applied soft-constrained and
hard-constrained network emulators to atmospheric data. Their results showed
that enforcing constraints, whether soft or hard, can systematically reduce
model error, but the hard-constrained model is free of an accuracy-constraint
trade-off. In addition, Daw et al. (2020) developed constrained long short-
term memory models to emulate lake water temperature dynamics. Their outcomes
reflect the same pattern observed in Beucler et al. (2021).

###  2.2 Deep Learning for Climate Downscaling

Statistical downscaling of climate data using deep learning has attracted much
attention over the last few years. Given the popularity of convolutional
neural networks (Dong et al., 2015, CNNs) and generative adversarial networks
(Goodfellow et al., 2014, GANs) for super-resolution of natural images, they
have become popular architecture choices for downscaling. Chen et al. (2022);
Watson et al. (2020); Chaudhuri and Robertson (2020) used CNNs and GANs to
downscale precipitation fields, while Harder et al. (2023) used CNNs and GANs
to downscale other quantities such as water content and temperature. So far,
climate downscaling works have mainly focused on increasing the resolution in
either spatial or temporal dimensions. Recently, Harder et al. (2022b)
introduced a new spatiotemporal downscaling model (increasing resolution in
both spatial and temporal dimensions), which stacks Deep Voxel Flow model (Liu
et al., 2017) and ConvGRU network (Ballas et al., 2015). It is able to
generate accurate and reliable high-resolution outputs when a customized
physics constraint layer is applied.

###  2.3 Fourier Neural Operators

In a classic deep learning setting, a neural network is trained to approximate
a function that forms a mapping between finite-dimensional spaces. Recent work
by Li et al. (2020) generalized neural networks to neural operators, which can
learn mappings between two infinite dimensional spaces (e.g., function
spaces)—while keeping a finite set of parameters to define the neural
architecture. They are typically trained in a supervised fashion to solve
parameterized PDEs and demonstrate comparable performance to numerical solvers
(Kovachki et al., 2023). Fourier Neural Operators (FNOs) (Li et al., 2021)
extended neural operators to enable feature transformations with parameters
defined in Fourier domain, resulting in an expressive and efficient
architecture. FNOs became the first neural operator model to successfully
learn a convergent solution operator for the Navier-Stokes equations in a
turbulent regime.

##  3 Methodology

###  3.1 Problem Setup

Consider low-resolution input $\mathbf{a}\in\mathbb{R}^{d_{a}}$ and high-
resolution output $\mathbf{b}\in\mathbb{R}^{d_{b}}$ with $d_{a}<d_{b}$.
Traditional neural network downscaling models define a mapping
$f:\mathbb{R}^{d_{a}}\rightarrow\mathbb{R}^{d_{b}}$ from low-resolution input
$\mathbf{a}$ to high-resolution output $\mathbf{b}$. This formulation induces
a limitation where the downscaled output resolution is fixed to be $d_{b}$. We
propose the following formulation to relax this limitation to achieve
arbitrary resolution downscaling.

Instead of looking for a mapping between two finite-dimensional spaces, our
methodology learns a mapping from a finite-dimensional space to an infinite-
dimensional space. Namely, this mapping takes in low-resolution input
$\mathbf{a}\in\mathbb{R}^{d_{a}}$ and outputs a function
$\mathbf{u}\in\mathcal{U}$ of which a high-resolution observation $\mathbf{b}$
is a discretization. We denote this mapping as:
$G^{\dagger}:\mathbb{R}^{d_{a}}\rightarrow\mathcal{U}$, where
$\mathcal{U}=\mathcal{U}(D;\mathbb{R}^{d_{u}})$ is a Banach space of functions
taking values in $\mathbb{R}^{d_{u}}$ at each point from a bounded open set
$D\subset\mathbb{R}^{d}$. $D$ can be viewed as a $d$-dimensional hypercube. As
a result, arbitrarily high-resolution outputs can be obtained by evaluating
$\mathbf{u}$ at arbitrarily many points from $D$.

Suppose we have observations $\\{\mathbf{a}_{j},\mathbf{u}_{j}\\}_{j=1}^{N}$,
where $\mathbf{a}_{j}$ is an i.i.d. low-resolution sample and
$\mathbf{u}_{j}=G^{\dagger}(\mathbf{a}_{j})$, function interpolating the high-
resolution counterpart, is possibly corrupted with some random noise. We aim
to construct a parametric map as follows to approximate $G^{\dagger}$:

$G:\mathbb{R}^{d_{a}}\times\Theta\rightarrow\mathcal{U}\;\;\;\;\;\;\text{or
equivalently,}\;\;\;\;\;\;G_{\theta}:\mathbb{R}^{d_{a}}\rightarrow\mathcal{U},\;\theta\in\Theta,$
(1)

where $\Theta$ is a finite-dimensional parameter space. We aim to find a
${\theta}^{\dagger}\in\Theta$ such that
$G(\mathbf{a},{\theta}^{\dagger})=G_{\theta^{\dagger}}(\mathbf{a})$ is close
to $G^{\dagger}(\mathbf{a})$, which can be formulated as an optimization
problem:

$\theta^{\dagger}=\arg\min_{\theta\in\Theta}\mathbb{E}_{\mathbf{a}}[C(G(\mathbf{a},\theta),G^{\dagger}(\mathbf{a}))],$
(2)

where $C:\mathcal{U}\times\mathcal{U}\rightarrow\mathbb{R}$ is a cost
functional measuring the distance in $\mathcal{U}$. In the following
experiments, we take $\mathbf{a}_{j}$ as a single channel low resolution image
and $\mathbf{u}_{j}\in\mathcal{U}((0,1)^{2};\mathbb{R})$ as a function
interpolating its high resolution counterpart.

Note that our data $\mathbf{u}_{j}$ are functions. In practice, to work with
$\mathbf{u}_{j}$ numerically, we assume access to point-wise evaluations of
it, which is denoted as $u_{j}$. It is generated by a discretization operator
$T$ applied to $\mathbf{u}_{j}$. Formally,

$u_{j}=T(\mathbf{u}_{j},\mathbb{D})=\\{\mathbf{u}_{j}(x_{1}),\dots,\mathbf{u}_{j}(x_{n})\\},$
(3)

where $\mathbb{D}=\\{x_{1},\dots,x_{n}\\}\subset D$ is a $n$-point
discretization of the domain $D$.

###  3.2 Implementation

Unlike neural operators which map between function spaces, the downscaling
model $G_{\theta}$ defined in the previous section learns a mapping from a
vector space to a function space. To achieve this transformation, a
discretization inversion operator (mapping from a vector to a function)
denoted as $T^{-1}$ is applied inside $G_{\theta}$. A neural network (mapping
between vectors) denoted as $f_{\theta}$ and a neural operator (mapping
between functions) denoted as $\mathcal{F}_{\theta}$ are also stacked before
and after the discretization inversion operator to increase the capacity of
the whole model. Therefore, we construct $G_{\theta}$ as follows:

$G_{\theta}(\mathbf{a}):=\mathcal{F}_{\theta}(T^{-1}(f_{\theta}(\mathbf{a}))).$
(4)

$f_{\theta}:\mathbb{R}^{d_{a}}\rightarrow\mathbb{R}^{d}$ is a vector-valued
function parameterized by a neural network; the discretization inversion
operator $T^{-1}:\mathbb{R}^{d}\rightarrow\mathcal{E}(D;\mathbb{R}^{d_{e}})$
is implemented as an interpolation scheme, which interpolates the output of
$f_{\theta}$ as a function $\mathbf{e}\in\mathcal{E}$ over domain $D$; and
$\mathcal{F}_{\theta}:\mathcal{E}\rightarrow\mathcal{U}$ is a functional
operator parameterized by a neural operator (Li et al., 2020). Here $T^{-1}$
can be a very simple interpolation scheme (e.g., linear interpolation) without
hurting the expressiveness of the overall model $G_{\theta}$. There are two
reasons for it. First, $f_{\theta}$ is able to learn a high-dimensional
embedding such that a simple interpolation of it would retain high
expressiveness for the target with lower dimensionality. Second,
$\mathcal{F}_{\theta}$ can learn a highly non-linear operator to apply
complicated transformations to the interpolated function
$\mathbf{e}=T^{-1}(f_{\theta}(\mathbf{a}))$ despite of the simple components
of $\mathcal{F}_{\theta}$ (Li et al., 2021). During inference, by evaluating
$\mathbf{e}$ at a specific resolution over a domain $D$, we can obtain the
downscaled output at any desired resolution.

In this work, $f_{\theta}$ is represented by a residual convolutional network
inspired by the generator architecture of a widely used super-resolution GAN
(Wang et al., 2018); an FNO is implemented for $\mathcal{F}_{\theta}$; and
bicubic interpolation is used as $T^{-1}$. Figure 1(a) shows an illustration
of the overall structure of our proposed downscaling FNO (DFNO) model, denoted
by $G_{\theta}$. The detailed architecture of neural network $f_{\theta}$ is
pictured in Figure 1(b). For the FNO $\mathcal{F}_{\theta}$, we use the same
architecture as described in Li et al. (2021).

Figure 1: The upper panel shows the overall structure of our Fourier neural
operator downscaling model denoted by $G_{\theta}$. The low-resolution input
$\mathbf{a}$ goes through a neural network $f_{\theta}$ and a discretization
inversion operator $T^{-1}$. Then an embedding function $\mathbf{e}(\cdot)$
over domain $D$ is returned. Finally, a neural operator $\mathcal{F}_{\theta}$
takes in $\mathbf{e}(\cdot)$ and outputs the target function
$\mathbf{u}(\cdot)$ which interpolates the high-resolution observation of
$\mathbf{a}$. The lower panel shows the detailed architecture of $f_{\theta}$,
which starts and ends with a convolutional layer, sandwiching a series of
convolutional residual blocks.

##  4 Experiments

###  4.1 Downscaling PDE Data

In order to evaluate the performance of our model to downscale PDE data, we
used a dataset solving the 2D Navier-Stokes equation for a viscous and
incompressible fluid in vorticity form (Li et al., 2021, Section 5.3). The
equation was numerically solved $10000$ times at resolution $64\times 64$ with
randomly sampled initial conditions. Each solution was integrated for $50$
time steps with a viscosity of $\text{10}^{-4}$. Out of $10000$ solutions,
$7000$, $2000$, and $1000$ solutions were sampled as train, validation, and
test sets, respectively. The solutions at each time step were then downsampled
via average pooling to resolutions $32\times 32$ and $16\times 16$. Our PDE
downscaling dataset consists of the solutions along with the downsampled
versions.

Following implementation details specified in Section 3.2, we constructed our
DFNO model and trained it on the PDE downscaling dataset with upsampling
factor 2 (i.e., $16\times 16\rightarrow 32\times 32$), and then evaluated it
at both 2 times (learned) and 4 times (zero-shot) downscaling.

As baselines for comparison, we trained two CNN (CNN-2 and CNN-4) and two GAN
(GAN-2 and GAN-4) downscaling models with pre-defined upsampling factors 2 and
4. The network architectures follow the design in the paper by Harder et al.
(2022b). The baseline models were trained on datasets of their corresponding
upsampling factors, and their outputs were then adjusted to achieve the
desired resolution for evaluation. Downscaling outputs from 2 times models
(CNN-2 and GAN-2) increase their resoultion to 4 times by model recursion and
bicubic interpolation. Correspondingly, downscaling outputs from 4 times
models (CNN-4 and GAN-4) decrease their resolution to 2 times by average
pooling and bicubic interpolation. As an additional simple, non-deep learning
baseline, we also considered bicubic interpolation (de Boor, 1962) to the
target resolution.

For reliable usage of downscaled results in downstream tasks, it is important
for results to be both close to the ground truth and physically consistent.
Harder et al. (2022b) showed that a softmax constraint layer can effectively
enforce conservation laws in neural networks for downscaling, without
decreasing accuracy. Thus, we conducted another set of experiments where all
aforementioned models include an additional softmax constraint layer at its
end to generate physically consistent outputs.

To evaluate each downscaling model, we computed the improvement with respect
to the unconstrained bicubic baseline. In particular, in the case of error
metrics, that is the mean squared error (MSE) and mean absolute error (MAE),
the improvement was computed as the error of the bicubic baseline divided by
the error of the evaluated model. In the case of the peak signal-to-noise
ratio (PSNR) and the structural similarity index measure (SSIM), the
improvement was computed as $100\times(M-B)/B$, where $B$ is the result of the
bicubic baseline and $M$ is the result of the evaluated model. These derived
relative results facilitate the comparison across models. These results are
summarized visually in Figure 2 and we provide the evaluation numerical
details in Tables 2, 4, 3, and 5.

In the unconstrained cases, DFNO shows dominant performance over all baseline
models in all evaluation metrics for 2 times downscaling on which it was
trained. This performance advantage persists when the DFNO model trained on 2
times downscaling is asked to zero-shot generalize to 4 times downscaling,
where it outperforms models directly trained on the 4 times downscaling
dataset such as CNN-4 and GAN-4. After the constraint layer is applied, DFNO’s
skill is further boosted for both 2 times and 4 times downscaling. It is
consistent with the conclusion by Harder et al. (2022b) that training networks
with the constraint layer can introduce an inductive bias, helping networks
give more accurate downscaling results. However, note that in the zero-shot
downscaling cases where bicubic interpolation is used to adjust network output
resolution (i.e., 4 times downscaling with CNN-2, 2 times downscaling with
CNN-4, 4 times downscaling with GAN-2, and 2 times downscaling with GAN-4),
the constraint layer generally degrades model performance. This is probably
because these networks are not trained to adapt to the renormalization
operation inside the constraint layer with a different upsampling factor.

One PDE solution downscaling example by our constrained DFNO model is
presented in Figure 3. The top row shows the result of input reconstruction (1
time downscaling). Because of the softmax constraint layer, DFNO trivially
reconstructs the exact input because the conservation law enforces the output
to equal the input when the upsampling factor is 1. Rows 2 and 3 illustrate 2
times (learned) and 4 times (zero-shot) downscaling results by DFNO. In both
cases, the downscaled outputs (column 1) are very close to the ground truth
(column 2), and the difference (column 3) is minor and negligible with values
roughly one order of magnitude lower than the ground truth values.

Figure 2: Metrics for downscaling models applied to the PDE dataset.
Downscaling models CNN-2 (CNN-4) and GAN-2 (GAN-4) are trained with 2 times (4
times) downscaling data; the DFNO model is only trained with 2 times
downscaling data. Each downscaling model is evaluated on both 2 times and 4
times downscaling. The 2 times downscaling outputs by CNN-2 and GAN-2 increase
their resolution to 4 times through model recursion and bicubic interpolation.
The 4 times downscaling outputs by CNN-4 and GAN-4 decrease their resolution
to 2 times through average pooling and bicubic interpolation. Square (dot)
denotes constrained (unconstrained) models. The metric mean and confidence
interval from 3 runs are shown relatively to unconstrained bicubic
interpolation. Model performance is evaluated by comparing marks of the same
upsampling factor: cold colors for 2 times and warm colors for 4 times. Metric
numerics and more details can be found in Tables 2, 4, 3, and 5. Figure 3:
This figure shows the downscaling performance of our DFNO model with softmax
constraint layer on the PDE solution data. The DFNO model was trained with 2
times downscaling data, then evaluated at 1 time (row 1), 2 times (row 2), and
4 times (row 3) downscaling. Column 1 shows the outputs from our DFNO model;
column 2 is the numerical solution ground truth; and the difference (one order
of magnitude lower than the ground truth values) between truth and prediction
is presented in column 3.

###  4.2 Downscaling ERA5 Climate Data

The ERA5 climate and weather dataset (Hersbach et al., 2020) is a reanalysis
product from the European Center for Medium-Range Weather Forecasts (ECMWF)
that combines model data with worldwide observations. The observations are
used as boundary conditions for numerical models that then predict various
atmospheric variables. ERA5 is available as global hourly data with a
$0.25^{\circ}\times 0.25^{\circ}$ resolution, which is roughly
$25\leavevmode\nobreak\ \mbox{km}$ per pixel. It covers all years starting
from 1950.

For this work, the quantity we focus on is the total column water that
describes the vertical integral of the total amount of atmospheric water
content, including water vapor, cloud water, and cloud ice but not
precipitation. At each time step, we extract a random $128\times 128$ patch
from the global water content field of size $721\times 1440$. There are
roughly 60,000 time steps available in total. From these, 40,000 patches are
randomly sampled for training and 10,000 for each validation and testing. The
low-resolution counterparts are created by average pooling on high-resolution
samples following the standard practice as in Serifi et al. (2021); Leinonen
et al. (2021). It results in low-resolution samples of sizes $32\times 32$ and
$64\times 64$. This operation is physically sound, considering that
conservation of water content means that the water content (density per
squared meter) described in a low-resolution pixel should be equal to the
average of its corresponding high-resolution pixels.

As in the previous section, a DFNO model is trained with 2 times downscaling
data and tested at 1 times, 2 times, and 4 times downscaling. Its performance
is also compared against two CNN and two GAN downscaling models of upsampling
factors 2 and 4. To enforce conservation law, a separate set of experiments
are conducted with the softmax constraint layer applied. The downscaling
performance of all models is collected in Tables 6 and 8 (without constraint
layer) and Tables 7 and 9 (with constraint layer), and we provide a
visualization of the relative improvement with respect to the bicubic baseline
in Figure 4.

When the constraint layer is not applied, in learned (2 times) downscaling, we
find that DFNO has the highest skill among all baseline models in all
evaluation metrics. For zero-shot (4 times) downscaling, DFNO has an MAE score
slightly worse than baseline CNN-2, CNN-4 and GAN-2 models but shows
performance dominance in all the other metrics. Better performance in terms of
MSE than MAE means the DFNO prediction errors mostly concentrate at values
with magnitude less than 1. It is likely due to the fact that our DFNO is
trained with MSE as the loss function, which is more sensitive to errors with
large magnitude. After applying the constraint layer, DFNO performance in both
learned and zero-shot downscaling is boosted showing performance dominance for
all metrics.

Figure 4: Metrics for downscaling models applied to the ERA5 dataset.
Downscaling models CNN-2 (CNN-4) and GAN-2 (GAN-4) are trained with 2 times (4
times) downscaling data; the DFNO model is only trained with 2 times
downscaling data. Each downscaling model is evaluated on both 2 times and 4
times downscaling. The 2 times downscaling outputs by CNN-2 and GAN-2 increase
their resolution to 4 times through model recursion and bicubic interpolation.
The 4 times downscaling outputs by CNN-4 and GAN-4 decrease their resolution
to 2 times through average pooling and bicubic interpolation. Square (dot)
denotes constrained (unconstrained) models. The metric mean and confidence
interval from 3 runs are shown relatively to unconstrained bicubic
interpolation. Model performance is evaluated by comparing marks of the same
upsampling factor: cold colors for 2 times and warm colors for 4 times. Metric
numerics and more details can be found in Tables 6, 8, 7, and 9. Figure 5:
This figure shows the downscaling performance of our DFNO model with softmax
constraint layer on ERA5 water content data. The DFNO model was trained with 2
times downscaling data, then evaluated at 1 time (row 1), 2 times (row 2), and
4 times (row 3) downscaling. Column 1 shows the outputs from our DFNO model;
column 2 is the ground truth; the difference between truth and prediction is
presented in column 3.

Figure 5 illustrates a case study on constrained DFNO downscaling ERA5 water
content data. The softmax constraint layer helps DFNO reconstruct input
perfectly (row 1). The 2 times downscaled (row 2) and 4 times downscaled (row
3) outputs are visually close to the ground truth (column 2) and with rather
high perceptual quality as validated by quantitative metric scores in Tables 7
and 9. However, the error in column 3 is not as small as in the case of PDE
solution downscaling (Figure 3). It is not surprising as our model is intended
for PDE based climate simulation data downscaling rather than observational
climate data downscaling; the FNO inside our model applies transformations on
a truncated Fourier series, so it is naturally easier for it to model
simulation data which have a more succinct representation in Fourier basis
than observational data.

###  4.3 Downscaling for PDE Integration

This section considers the use of DFNO in integrating PDEs at high resolution
(i.e., generating high resolution PDE solutions). There has been increasing
interest in the use of data-driven deep learning-based methods to predict PDE
solutions autoregressively (Li et al., 2021), and the Fourier neural operator
was introduced as a state-of-the-art approach in this regard. Here, we show
that the DFNO paradigm has the potential to significantly improve upon the
standard FNO approach. Namely, we assume that we have access to an accurate
_low-resolution_ PDE solver, then use the DFNO to downscale the solution to
higher resolution. Having a low-resolution PDE solution is a plausible
assumption, since traditional numerical solvers are prohibitively time-
intensive at high resolution but can be very cheap to run at low resolution.

Here we consider the Navier-Stokes equation as in Section 4.1. In the previous
two sections, we have seen that the constrained DFNO outperforms the
unconstrained DFNO; as a result, we here use only the constrained model. We
train two different DFNO models, using 2 times ($16\times 16\rightarrow
32\times 32$) and 4 times ($16\times 16\rightarrow 64\times 64$) PDE
downscaling data, respectively, and denoted as DFNO-2 and DFNO-4. We compare
our approach against the standard FNO method, which predicts a solution one
time step forward based on the solution at the previous ten time steps; two
FNO models are trained with solution data at resolution $32\times 32$ and
$64\times 64$, and are denoted as FFNO-32 and FFNO-64. Becasue FFNO-32 and
FFNO-64 are resolution invariant, both of them are tested solving the Navier-
Stokes equation at resolutions $32\times 32$ and $64\times 64$. In the end,
all four models are evaluated with generated PDE solutions at resolutions
$32\times 32$ and $64\times 64$.

The solutions generated by DFNO and FFNO models are compared against ground
truth numerical solutions, and the performance is summarized in Table 1.
Overall, DFNO models show a significant performance advantage over FFNO
models. Comparing between DFNO models, it is not surprising that zero-shot
downscaling is still not as good as learned downscaling. To evaluate DFNO and
FFNO performance visually, solution examples generated by FFNO-64 and DFNO-2
at resolution $64\times 64$ for five consecutive time steps are presented in
Figures 6 and 7. The generated solutions (column 1) are very close to
numerical solutions (column 2) for both models. On the other hand, even though
DFNO-2 zero-shot results are compared against FFNO-64 learned results, the
error magnitude by DFNO-2 (column 3) is much less than that by FFNO-64.
Results from both quantitative metric scores and visual illustration
demonstrate that downscaling low-resolution solutions from numerical solvers
gives better accuracy than generating by data-driven high-resolution solvers,
that is, inputting low-resolution solutions as guidance makes it much easier
to generate realistic high-resolution solutions.

Table 1: This table compares two ways of solving the Navier-Stokes equation at high resolution concerning mean squared error (MSE) and mean absolute error (MAE). First: solve the equation numerically at low resolution ($16\times 16$), then downscale the solution to $32\times 32$ and $64\times 64$ by constrained DFNO models. Second: use data-driven FFNO models to auto-regressively predict solutions at resolution $32\times 32$ and $64\times 64$. Metric | Resolution | DFNO-2 | DFNO-4 | FFNO-32 | FFNO-64
---|---|---|---|---|---
MSE | $32\times 32$ | 0.0004 | 0.0012 | 0.0101 | 0.0113
MSE | $64\times 64$ | 0.0018 | 0.0007 | 0.0136 | 0.0118
MAE | $32\times 32$ | 0.0124 | 0.0208 | 0.0677 | 0.0725
MAE | $64\times 64$ | 0.0246 | 0.0168 | 0.0788 | 0.0739
Figure 6: This figure shows Navier-Stokes equation solution $(64\times 64)$ at
five consecutive time steps (row 1 to row 5). The solution is generated by
FFNO-64, a forward solution prediction model trained on a solution dataset of
resolution $64\times 64$. Column 1 shows FFNO-64 predicted solution; column 2
is the numerical solution ground truth; column 3 shows the difference between
column 1 and column 2. Figure 7: Similar to Figure 6 but the solution is
generated by DFNO-2. It is a constrained DFNO model trained on solution
downscaling data from $16\times 16$ to $32\times 32$. It performs zero-shot
downscaling on a solution from $16\times 16$ to $64\times 64$.

##  5 Conclusion

In this work, we introduce the first arbitrary resolution downscaling model
for climate data. This model takes in a low-resolution sample and outputs a
function that interpolates the observed high-resolution counterpart. The low-
resolution input is downscaled to an arbitrarily high resolution by evaluating
the output function at discrete points. This model consists of three
components: neural network, discretization inversion operator, and neural
operator. They are implemented respectively as a residual convolutional
network, bicubic interpolation, and a Fourier neural operator.

Our model is evaluated on a Navier-Stokes equation solution dataset and an
ERA5 reanalysis water content dataset. It improves downscaling performance on
both datasets significantly relative to state-of-the-art CNN and GAN super-
resolution methods. It also zero-shot generalizes to higher upsampling
factors, outperforming models directly trained on those factors. Our model’s
performance is further boosted when a softmax constraint layer is applied to
enforce conservation laws. Finally, we compare two ways to integrate PDEs at
high resolution. Combining our downscaling model with a low-resolution
numerical solver, the downscaled solution has superior accuracy to that of the
state-of-the-art high-resolution data-driven solver.

While our DFNO approach conveys a significant performance improvement across
all tasks, it demonstrates an even greater efficacy on climate simulation
(Navier-Stokes equation) data as compared to observational climate (total
water content) data. This may result from the fact that simulation data are
much smoother than observational data; that is, simulation data have a more
succinct representation in the Fourier basis than observational data.
Therefore, simulation data are easier to be captured by a Fourier neural
operator with a truncated Fourier series. It would be interesting to explore
how to modify our model to adapt to data without a succinct Fourier
representation so that its performance on observational climate data can be
further improved.

Acknowledgments and Disclosure of Funding

This work was supported in part by the Québec Ministère de l’Économie et de
l’Innovation, IBM, and the Canada CIFAR AI Chairs Program. The authors also
acknowledge material support from NVIDIA in the form of computational
resources, and are grateful for technical support from the Mila IDT team in
maintaining the Mila Compute Cluster.

## Appendix

Table 2: Downscaling performance on the PDE dataset in terms of mean squared error (MSE), mean absolute error (MAE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM). The best scores are highlighted in bold red, second best in bold blue. The DFNO model was trained on 2 times downscaling data, then tested on 1 time, 2 times, and 4 times downscaling. CNN-2 (GAN-2) and CNN-4 (GAN-2) represent convolutional (generative adversarial) downscaling models with predefined upsampling factors 2 and 4. They were trained on datasets of their corresponding upsampling factors, whose downscaling results are then downsampled or upsampled via bicubic interpolation to get desired resolution for evaluation. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0146 | 0.0057 | 0.0123 | 0.0056 | 0.0131 | 0.0000
MSE | $2\times$ | 0.0015 | 0.0042 | 0.0052 | 0.0044 | 0.0062 | 0.0252
MSE | $4\times$ | 0.0037 | 0.0093 | 0.0070 | 0.0095 | 0.0080 | 0.0350
MAE | $1\times$ | 0.0826 | 0.0524 | 0.0697 | 0.0520 | 0.0746 | 0.0000
MAE | $2\times$ | 0.0238 | 0.0397 | 0.0458 | 0.0424 | 0.0534 | 0.1027
MAE | $4\times$ | 0.0359 | 0.0579 | 0.0495 | 0.0601 | 0.0573 | 0.1150
PSNR | $1\times$ | 40.2750 | 44.3504 | 41.0302 | 44.4541 | 40.7810 | 154.0983
PSNR | $2\times$ | 50.2061 | 45.7778 | 44.8762 | 45.5806 | 44.2337 | 38.0326
PSNR | $4\times$ | 46.3361 | 42.4054 | 43.6083 | 42.3192 | 43.1123 | 36.6248
SSIM | $1\times$ | 0.9934 | 0.9968 | 0.9935 | 0.9963 | 0.9890 | 1.0000
SSIM | $2\times$ | 0.9981 | 0.9963 | 0.9952 | 0.9956 | 0.9917 | 0.9741
SSIM | $4\times$ | 0.9920 | 0.9842 | 0.9879 | 0.9835 | 0.9847 | 0.9335

Table 3: Similar to Table 2 but softmax constraint layer is applied to the output of each model. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MSE | $2\times$ | 0.0011 | 0.0038 | 0.0063 | 0.0038 | 0.0084 | 0.0365
MSE | $4\times$ | 0.0029 | 0.0217 | 0.0063 | 0.0228 | 0.0064 | 0.0517
MAE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MAE | $2\times$ | 0.0196 | 0.0363 | 0.0528 | 0.0365 | 0.0627 | 0.1241
MAE | $4\times$ | 0.0313 | 0.1032 | 0.0457 | 0.1058 | 0.0462 | 0.1431
PSNR | $1\times$ | 151.8861 | 153.3908 | 152.4238 | 153.3476 | 152.1304 | 152.4239
PSNR | $2\times$ | 51.8071 | 46.2719 | 44.2463 | 46.2266 | 43.0041 | 36.4336
PSNR | $4\times$ | 47.4375 | 38.7146 | 44.1036 | 38.5096 | 44.0425 | 34.9377
SSIM | $1\times$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
SSIM | $2\times$ | 0.9987 | 0.9969 | 0.9942 | 0.9969 | 0.9920 | 0.9659
SSIM | $4\times$ | 0.9937 | 0.9605 | 0.9894 | 0.9583 | 0.9892 | 0.9108

Table 4: Downscaling performance on the PDE dataset in terms of mean squared error (MSE), mean absolute error (MAE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM). The best scores are highlighted in bold red, second best in bold blue. The DFNO model was trained on 2 times downscaling data, then tested on 1 time, 2 times, and 4 times downscaling. CNN-2 (GAN-2) and CNN-4 (GAN-2) represent convolutional (generative adversarial) downscaling models with predefined upsampling factors 2 and 4. They were trained on datasets of their corresponding upsampling factors, whose downscaling results are then downsampled (upsampled) via average pooling (model recursion) to get desired resolution for evaluation. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0146 | 0.0002 | 0.0002 | 0.0004 | 0.0011 | 0.0000
MSE | $2\times$ | 0.0015 | 0.0042 | 0.0042 | 0.0044 | 0.0051 | 0.0252
MSE | $4\times$ | 0.0037 | 0.0076 | 0.0070 | 0.0083 | 0.0080 | 0.0350
MAE | $1\times$ | 0.0826 | 0.0097 | 0.0098 | 0.0164 | 0.0237 | 0.0000
MAE | $2\times$ | 0.0238 | 0.0397 | 0.0394 | 0.0424 | 0.0477 | 0.1027
MAE | $4\times$ | 0.0359 | 0.0535 | 0.0495 | 0.0600 | 0.0573 | 0.1150
PSNR | $1\times$ | 40.2750 | 60.8445 | 60.4059 | 56.6575 | 55.8089 | 154.0983
PSNR | $2\times$ | 50.2061 | 45.7778 | 45.8595 | 45.5806 | 45.1245 | 38.0326
PSNR | $4\times$ | 46.3361 | 43.2835 | 43.6083 | 42.8902 | 43.1123 | 36.6248
SSIM | $1\times$ | 0.9934 | 0.9996 | 0.9996 | 0.9989 | 0.9953 | 1.0000
SSIM | $2\times$ | 0.9981 | 0.9963 | 0.9963 | 0.9956 | 0.9928 | 0.9741
SSIM | $4\times$ | 0.9920 | 0.9868 | 0.9879 | 0.9849 | 0.9847 | 0.9335

Table 5: Similar to Table 4 but softmax constraint layer is applied to the output of each model. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MSE | $2\times$ | 0.0011 | 0.0038 | 0.0036 | 0.0038 | 0.0036 | 0.0365
MSE | $4\times$ | 0.0029 | 0.0067 | 0.0063 | 0.0067 | 0.0064 | 0.0517
MAE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MAE | $2\times$ | 0.0196 | 0.0363 | 0.0354 | 0.0365 | 0.0357 | 0.1241
MAE | $4\times$ | 0.0313 | 0.0474 | 0.0457 | 0.0478 | 0.0462 | 0.1431
PSNR | $1\times$ | 151.8861 | 149.9829 | 147.8327 | 149.3479 | 147.5434 | 152.4239
PSNR | $2\times$ | 51.8071 | 46.2719 | 46.5235 | 46.2266 | 46.4569 | 36.4336
PSNR | $4\times$ | 47.4375 | 43.8382 | 44.1036 | 43.7889 | 44.0425 | 34.9377
SSIM | $1\times$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
SSIM | $2\times$ | 0.9987 | 0.9969 | 0.9971 | 0.9969 | 0.9970 | 0.9659
SSIM | $4\times$ | 0.9937 | 0.9889 | 0.9894 | 0.9887 | 0.9892 | 0.9108

Table 6: Downscaling performance on the ERA5 water content dataset in terms of mean squared error (MSE), mean absolute error (MAE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM). The best scores are highlighted in bold red, second best in bold blue. The DFNO model was trained on 2 times downscaling data, then tested on 1 time, 2 times, and 4 times downscaling. CNN-2 (GAN-2) and CNN-4 (GAN-2) represent convolutional (generative adversarial) downscaling models with predefined upsampling factors 2 and 4. They were trained on datasets of their corresponding upsampling factors, whose downscaling results are then downsampled or upsampled via bicubic interpolation to get desired resolution for evaluation. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.2140 | 0.0940 | 0.1566 | 0.0930 | 0.1752 | 0.0000
MSE | $2\times$ | 0.2063 | 0.2488 | 0.2677 | 0.2474 | 0.2815 | 0.4201
MSE | $4\times$ | 0.3628 | 0.3870 | 0.3851 | 0.3853 | 0.3970 | 0.5954
MAE | $1\times$ | 0.2896 | 0.1737 | 0.2149 | 0.1731 | 0.2439 | 0.0000
MAE | $2\times$ | 0.2392 | 0.2541 | 0.2668 | 0.2541 | 0.2920 | 0.3380
MAE | $4\times$ | 0.3067 | 0.3023 | 0.3010 | 0.3022 | 0.3251 | 0.3838
PSNR | $1\times$ | 46.9630 | 50.5294 | 48.3152 | 50.5795 | 47.8863 | 173.5160
PSNR | $2\times$ | 48.1002 | 47.2860 | 46.9688 | 47.3110 | 46.7714 | 45.0115
PSNR | $4\times$ | 46.0154 | 45.7349 | 45.7560 | 45.7535 | 45.6334 | 43.8633
SSIM | $1\times$ | 0.9964 | 0.9982 | 0.9971 | 0.9982 | 0.9971 | 1.0000
SSIM | $2\times$ | 0.9941 | 0.9933 | 0.9933 | 0.9934 | 0.9932 | 0.9891
SSIM | $4\times$ | 0.9895 | 0.9882 | 0.9886 | 0.9884 | 0.9887 | 0.9835

Table 7: Similar to Table 6 but softmax constraint layer is applied to the output of each model. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MSE | $2\times$ | 0.1696 | 0.2181 | 0.2896 | 0.2181 | 0.2964 | 0.8314
MSE | $4\times$ | 0.2779 | 0.6054 | 0.3334 | 0.6118 | 0.3355 | 1.1552
MAE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MAE | $2\times$ | 0.2250 | 0.2427 | 0.3055 | 0.2422 | 0.3116 | 0.5318
MAE | $4\times$ | 0.2768 | 0.4383 | 0.2837 | 0.4386 | 0.2851 | 0.5950
PSNR | $1\times$ | 164.1793 | 170.2039 | 166.2301 | 169.6977 | 165.4083 | 161.0459
PSNR | $2\times$ | 48.9508 | 47.8584 | 46.6269 | 47.8584 | 46.5268 | 42.0471
PSNR | $4\times$ | 47.1723 | 43.7915 | 46.3820 | 43.7464 | 46.3549 | 40.9850
SSIM | $1\times$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
SSIM | $2\times$ | 0.9952 | 0.9937 | 0.9919 | 0.9938 | 0.9917 | 0.9778
SSIM | $4\times$ | 0.9910 | 0.9792 | 0.9893 | 0.9793 | 0.9892 | 0.9639

Table 8: Downscaling performance on the ERA5 water content dataset in terms of mean squared error (MSE), mean absolute error (MAE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM). The best scores are highlighted in bold red, second best in bold blue. The DFNO model was trained on 2 times downscaling data, then tested on 1 time, 2 times, and 4 times downscaling. CNN-2 (GAN-2) and CNN-4 (GAN-2) represent convolutional (generative adversarial) downscaling models with predefined upsampling factors 2 and 4. They were trained on datasets of their corresponding upsampling factors, whose downscaling results are then downsampled (upsampled) via average pooling (model recursion) to get desired resolution for evaluation. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.2140 | 0.0030 | 0.0046 | 0.0028 | 0.0243 | 0.0000
MSE | $2\times$ | 0.2063 | 0.2488 | 0.2573 | 0.2474 | 0.2713 | 0.4201
MSE | $4\times$ | 0.3628 | 0.3757 | 0.3851 | 0.3737 | 0.3970 | 0.5954
MAE | $1\times$ | 0.2896 | 0.0345 | 0.0440 | 0.0339 | 0.1087 | 0.0000
MAE | $2\times$ | 0.2392 | 0.2541 | 0.2592 | 0.2541 | 0.2852 | 0.3380
MAE | $4\times$ | 0.3067 | 0.2982 | 0.3010 | 0.2987 | 0.3251 | 0.3838
PSNR | $1\times$ | 46.9630 | 65.5137 | 63.8092 | 65.8671 | 59.5288 | 173.5160
PSNR | $2\times$ | 48.1002 | 47.2860 | 47.1402 | 47.3110 | 46.9310 | 45.0115
PSNR | $4\times$ | 46.0154 | 45.8635 | 45.7560 | 45.8862 | 45.6334 | 43.8633
SSIM | $1\times$ | 0.9964 | 1.0000 | 0.9999 | 1.0000 | 0.9998 | 1.0000
SSIM | $2\times$ | 0.9941 | 0.9933 | 0.9932 | 0.9934 | 0.9932 | 0.9891
SSIM | $4\times$ | 0.9895 | 0.9890 | 0.9886 | 0.9892 | 0.9887 | 0.9835

Table 9: Similar to Table 8 but softmax constraint layer is applied to the output of each model. Metric | Factor | DFNO | CNN-2 | CNN-4 | GAN-2 | GAN-4 | Bicubic
---|---|---|---|---|---|---|---
MSE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MSE | $2\times$ | 0.1696 | 0.2181 | 0.2188 | 0.2181 | 0.2202 | 0.8314
MSE | $4\times$ | 0.2779 | 0.3347 | 0.3334 | 0.3343 | 0.3355 | 1.1552
MAE | $1\times$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
MAE | $2\times$ | 0.2250 | 0.2427 | 0.2426 | 0.2422 | 0.2436 | 0.5318
MAE | $4\times$ | 0.2768 | 0.2844 | 0.2837 | 0.2833 | 0.2851 | 0.5950
PSNR | $1\times$ | 164.1793 | 155.5239 | 153.5080 | 155.0793 | 153.3323 | 161.0459
PSNR | $2\times$ | 48.9508 | 47.8584 | 47.8454 | 47.8584 | 47.8165 | 42.0471
PSNR | $4\times$ | 47.1723 | 46.3649 | 46.3820 | 46.3708 | 46.3549 | 40.9850
SSIM | $1\times$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
SSIM | $2\times$ | 0.9952 | 0.9937 | 0.9937 | 0.9938 | 0.9937 | 0.9778
SSIM | $4\times$ | 0.9910 | 0.9892 | 0.9893 | 0.9893 | 0.9892 | 0.9639

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