# Climate Intervention Analysis using AI Model Guided by Statistical Physics
Principles

Soo Kyung Kim  Palo Alto Research Center  Kalai Ramea  Palo Alto Research
Center  Salva Rühling Cachay  University of California, San Diego  Haruki
Hirasawa  University of Victoria  Subhashis Hazarika  Palo Alto Research
Center  Dipti Hingmire  University of Victoria  Peetak Mitra  Excarta  Philip
J. Rasch  University of Washington  Hansi A. Singh  University of Victoria

###### Abstract

The availability of training data remains a significant obstacle for the
implementation of machine learning in scientific applications. In particular,
estimating how a system might respond to external forcings or perturbations
requires specialized labeled data or targeted simulations, which may be
computationally intensive to generate at scale. In this study, we propose a
novel solution to this challenge by utilizing a principle from statistical
physics known as the Fluctuation-Dissipation Theorem (FDT) to discover
knowledge using an AI model that can rapidly produce scenarios for different
external forcings. By leveraging FDT, we are able to extract information
encoded in a large dataset produced by Earth System Models, which includes
8250 years of internal climate fluctuations, to estimate the climate system’s
response to forcings. Our model, AiBEDO, is capable of capturing the complex,
multi-timescale effects of radiation perturbations on global and regional
surface climate, allowing for a substantial acceleration of the exploration of
the impacts of spatially-heterogenous climate forcers. To demonstrate the
utility of AiBEDO, we use the example of a climate intervention technique
called Marine Cloud Brightening, with the ultimate goal of optimizing the
spatial pattern of cloud brightening to achieve regional climate targets and
prevent known climate tipping points. While we showcase the effectiveness of
our approach in the context of climate science, it is generally applicable to
other scientific disciplines that are limited by the extensive computational
demands of domain simulation models. Source code of AiBEDO framework is made
available at https://github.com/kramea/kdd_aibedo. A sample dataset is made
available at https://doi.org/10.5281/zenodo.7597027. Additional data available
upon request.

###### keywords:

Fluctuation-Dissipation Theorem, Physics guided ML, Climate Change, Climate
Intervention modeling

##  1 Introduction

Machine learning has greatly benefited a variety of scientific disciplines
through its ability to extract valuable insights from large datasets. Notable
examples include the use of deep learning for numerical weather prediction
Bochenek and Ustrnul, (2022); Ren et al., (2021); Markovics and Mayer, (2022),
medical imaging diagnosis Shen et al., (2017); Razzak et al., (2018); Erickson
et al., (2017); Chan et al., (2020); Litjens et al., (2017), classification of
astronomical images Carrasco-Davis et al., (2019); Hansen and Takahashi,
(1984), and drug discovery with specific properties Chen et al., (2018);
Stephenson et al., (2019); Dara et al., (2022); Patel et al., (2020). These
applications involve direct application of machine learning algorithms on
large datasets in order to achieve a specific goal. Additionally, we have seen
a rise in the use of surrogate models, such as data-driven models trained on
simulation data, which can instantaneously reproduce output for a given set of
inputs, thereby significantly increasing process efficiency Bárkányi et al.,
(2021). Despite these achievements, these use cases still represent only a
small subset of problems studied by scientists.

A commonly researched topic in systems analysis, science, and engineering is
how a system responds to external forcing. This can encompass a wide range of
questions, such as how a bridge or aircraft responds to wind Matsumoto et al.,
(1995); Siringoringo and Fujino, (2012), how a certain material responds to
electrical conductivity or magnetic forces Reyne et al., (1987); Zukoski,
(1993), or how a financial market responds to geopolitical instability Hoque
and Zaidi, (2020); Sohag et al., (2022). As these inquiries often fall outside
of the training data distributions, they are typically addressed through
targeted simulation scenario runs. However, the complexity of the domain
simulation model can significantly impact the computational time required for
a single simulation run, ranging from a few hours to several months, which can
impede the ability to rapidly evaluate ”what-if” scenarios and gain insights
into the system’s response to various external forcings.

This limitation is particularly acute in climate science due to the high
computational cost of Earth System Models (ESM). ESMs are comprehensive
simulation models that integrate the physics of Earth’s myriad processes to
improve our understanding of the interactions between the atmosphere, oceans,
land surface, and ice. These models are used for studying future changes in
the Earth’s climate Flato, (2011); Giorgi and GAO, (2018). The related concept
of climate intervention refers to proposed methods aimed at reducing the
effects of climate change by manipulating the Earth’s environment Mah et al.,
(2020); Lee et al., (2019); Valentini and Rudisill, (2006). One key category
of intervention strategies is Solar Radiation Management Robock, (2020);
Nicholson et al., (2018), which involves reflecting sunlight back into space
to cool the Earth. There are two main approaches that have been proposed to
increase the reflection of sunlight on planetary scales: (a) Stratospheric
Aerosol Injection (SAI) Pope et al., (2012); Hulme, (2012) and (b) Marine
Cloud Brightening (MCB) Stjern et al., 2018b ; Latham et al., 2012a ; Latham
et al., 2012b . SAI involves the introduction of aerosol particles such as
sulphates into the stratosphere, while MCB involves the spraying of sea salt
aerosols into low clouds in the lower atmosphere. While SAI generally has
similar climate impacts irrespective of injection site, the effects of MCB are
regional and spatially heterogenous. Both techniques necessitate extensive
research and an in-depth understanding of their impacts on the Earth System,
such as identifying possible unintended consequences of interventions.
Moreover, because MCB is applied regionally, an immense number of different
intervention scenarios are possible, making it an ideal use case for a rapid
screening tool to evaluate efficacy before running costly ESM simulations.

Due to the complexity of modern ESMs, a single ESM simulation can take weeks
to months to obtain sufficient sample sizes for analysis. Thus, past
evaluations of SAI and MCB interventions have only considered a small number
of scenarios, which are often involve simplified interventions Kravitz et al.,
(2015). This is particularly true of MCB, where only a handful of cloud
perturbation scenarios have been considered Rasch et al., (2009); Jones et
al., (2009); Stjern et al., 2018a . As a result, a complete set of ESM
simulation runs for MCB that can be used to query a number of “what-if”
scenarios does not exist. Given the large number of possible combinations of
spatial and temporal cloud perturbations, studying them using ESMs is likely a
computationally intractable task. While machine learning has been used in a
variety of climate modeling contexts, such as improving and accelerating ESMs
through the use of data-driven and hybrid physics-informed surrogate models
Reichstein et al., (2019); Ardabili et al., (2020); Zhu et al., (2022); Lu and
Ricciuto, (2019); Weber et al., (2020), the problem of determining the outcome
of an external forcing (i.e., the outcome in a system due to an external cloud
perturbation in the case of MCB) remains a challenge as it is a complex
problem with limited data sets to train on. This makes direct application of
machine learning approaches difficult, and requires a creative solution.

We present AiBEDO, an AI model, rooted in an innovative application of the
Fluctuation-Dissipation Theorem (FDT). FDT is a fundamental principle in
statistical physics that states that the forced response of a system mirrors
its internal fluctuations Kubo, (1966). FDT has been successfully used in many
fields–brownian motion and drag in moving objects Tsekov, (2010); Sharma and
Patankar, (2004), protein folding in biological systems Hayashi and Takano,
(2007); Mo, (2006); Huang, (2008) and assessing climate sensitivity in ESMs
Leith, (1975); Gritsoun et al., (2002); Cionni et al., (2004); Dymnikov and
Gritsun, (2005); Gritsun and Branstator, (2007). Past implementations of FDT
in the scientific machine learning community mainly involve utilizing it to
ensure the consistency of the physical dynamics of a system. Recently, Lee et
al. (2021) Lee et al., (2021) apply FDT in machine learning to ensure
thermodynamic consistency while learning reversible and irreversible dynamics
of physical systems. A related but distinct study illustrates the conservation
of the stationary equilibrium of parameters of the ML model using FDT by
connecting measurable quantities and hyperparameters in the stochastic
gradient descent algorithm Yaida, (2018).

Our use of FDT in machine learning differs from the previous work and is
grounded in the fundamental application of learning from internal variability,
intrinsic chaotic fluctuations in a complex system, to estimate a forced
response. Performing this operation in the traditional modeling sense requires
large sample sizes and is limited by the use of linear response functions. Our
work illustrates a novel pathway to improve this baseline through the help of
AI models. To achieve this goal, we train the model on the climate response
over a wide range of natural fluctuations in the cloud radiative forcing
Ramanathan et al., (1989). The training data is derived from a large ensemble
of ESM simulations, amounting to a total of over 100,000 model months. The
model can then estimate responses to forced changes in radiation, obviating
the need to run many simulation runs and saving millions of core hours of
computing resources. We validate the model and verify that it can plausibly
project climate responses by comparing it with targeted ESM runs with regional
MCB-like forcing. Key contribution of our work are listed as follows:

  * •

We have showcased a novel method for developing a scenario analysis tool using
the principle of Fluctuation-Dissipation Theorem to develop an AI model
trained on noise to answer scientific queries requiring simulation runs with
external forcings.

  * •

Our AI framework maps the relationship between cloud perturbations and the
climate response, including large-scale circulation and regional climate
patterns. To the best of our knowledge, this is a pioneering application of AI
for scenario analysis of global-scale marine cloud brightening climate
interventions.

  * •

We have extensively evaluated the performance of AiBEDO against targeted
simulation runs of ESMs and show that our model can reproduce the results with
high fidelity, but three orders of magnitude faster than ESMs.

  * •

We have developed an interactive post-hoc analysis platform to examine the
results and facilitate rapid prototyping of MCB what-if scenarios for
downstream decision-making.

The remainder of the paper is organized as follows. Section 2 introduces the
problem statement and details of how FDT is applied in AiBEDO. In Section 3,
we see the details of different components of the MCB climate intervention
framework. In Section 4, we evaluate the performance of the data-driven
emulation component of AiBEDO against ESM runs, and we demonstrate the utility
of AiBEDO in an FDT-like approach to perform MCB intervention scenarios.
Finally, in Section 5, we discuss our results, limitations, and future
directions.

##  2 Problem Statement

The goal is to create a framework that one can use to rapidly generate MCB
intervention impacts for a given spatial and temporal extent of cloud
perturbations (external forcing). The problem will be addressed in two phases.
In the first phase, we develop an AI emulator to map relationships between a
set of designated input and output variables. This will involve the creation
of a series of mappings that are performed at different time-lagged intervals.
In the second phase, we sum them using an FDT operator to obtain a time-
integrated outcome, which estimates the regional impact of the external
forcing.

Variable | Description | Role in AiBEDO
---|---|---
cres | Net TOA shortwave cloud radiative effect | input
crel | Net TOA longwave cloud radiative effect | input
cresSurf | Net Surface shortwave cloud radiative effect | input
crelSurf | Net Surface longwave cloud radiative effect | input
netTOAcs | Net TOA clear-sky radiative flux | input
netSurfcs | Net surface clear-sky radiative flux plus all-sky surface heat flux | input
lsMask | Land fraction | input
ps | Surface pressure | output
tas | Surface air temperature | output
pr | Precipitation | output
Table 1: CESM2 LE variables used in AiBEDO. TOA - top of atmosphere. All
radiative and heat fluxes are positive down

###  2.1 Phase 1: AI Emulator of simultaneous and time-lagged mappings

Let us denote an input field of cloud and clearsky radiation anomalies at time
$t$ as $\vec{x}(t)$ and denote the corresponding output field of surface
climate anomalies after a time delay of $\tau$ as $\vec{y}(t+\tau)$. Our task
is then to develop a model $A_{\tau}$ to predict $y(t+\tau)$ from
$\vec{x}(t)$, i.e, $\vec{y}(t+\tau)=A_{\tau}(\vec{x}(t))$. Formally, the input
tensor $\delta\vec{x}(t)\in\mathbb{R}^{d\times c_{in}}$, and output tensor
$\vec{y}(t+\tau)\in\mathbb{R}^{d\times c_{out}}$ consist of 1-D climate data
of size $d$ with corresponding channels of size $c_{in}$ and $c_{out}$. Input
channels, $c_{in}$, are composed of 7 climate variables empirically selected
to achieve best performance of the model $A_{\tau}$, and output channels,
$c_{out}$, are composed of 7 climate variables. The list of climate variables
used for input and output is shown in Table 1.

###  2.2 Phase 2: Time-integrated output using Fluctuation-Dissipation theorem

One common method of applying FDT involves assuming the metrics of interest
have Gaussian statistics and constructing an FDT operator $\mathbf{L}$ using
the covariance matrix $\mathbf{C}(t)$ Cionni et al., (2004); Liu et al.,
(2018). The climate mean response $\delta\left<y\right>$ to a constant forcing
$\delta\vec{f}$ is then computed as

$\left<\delta
y\right>=\mathbf{L}^{-1}\delta\vec{f}=\left[\int_{0}^{\infty}\mathbf{C}(\tau)\mathbf{C}(0)^{-1}d\tau\right]\delta\vec{f}$
(1)

Noting that FDT is limited to the linear component of the climate response, we
use an AI model with the intention of capturing non-linear components of the
response and loosening some of the conditions required by classical FDT
(namely that the probability density function of the relevant climate
statistics must be Gaussian or quasi-Gaussian Cionni et al., (2004); Majda et
al., (2010)). Thus, replacing the linear FDT operator $\mathbf{L}$ with a set
of AiBEDO operators, $A_{\tau}$, at a series of time-lags ($\tau$), we
construct an estimate of the response as

$\left<{\delta\vec{y}(t)}\right>=\sum_{\tau=0}^{T_{max}}\frac{1}{N}\sum_{i=0}^{N}{\left(A_{\tau}(\vec{x_{i}}+\delta\vec{f}(t-\tau))-A_{\tau}(\vec{x_{i}})\right)}$
(2)

where $\vec{x_{i}}$ are randomly sampled internal fluctuations of the input
variables, $N$ is the number of samples of internal fluctuations used, and
$T_{max}$ is the upper lag limit set at the point when the response to a
perturbation approximately converges to noise. For testing purposes we use the
first 6 months and for the full lag integration we choose 48 months.

##  3 MCB Climate Intervention using AiBEDO: Data and Methods

The MCB climate intervention framework consists of three main components: (1)
datasets for training and verification, which includes a large ensemble of
Earth System Model output data; (2) AiBEDO, which serves as the framework’s
central component is an AI model incorporating data-driven models; (3) an
interactive visualization interface featuring dual functionality allowing
users to execute rapid “what-if” scenarios and enabling modelers to inspect
and provide explanations for the associated outcomes. A schematic of the
AiBEDO framework is shown in Figure 1.

Figure 1: Schematic view of MCB Climate Intervention framework

###  3.1 Data

Our training data consists of a subset of the most recent sixth generation of
Coupled Model Intercomparison (CMIP) ESM outputs. As FDT requires a large
amount of climate noise (i.e. internal variability or chaotic fluctuations)
for training, we use the Community Earth System Model 2 Large Ensemble
(CESM2-LE) as a source for internal climate noise (Rodgers et al.,, 2021, ;
Table 2). Specifically, we use the 50-member ensemble of historical
simulations with smoothed biomass burning emission between 1997 and 2014 at
nominal 1 degree spatial resolution and at a monthly temporal resolution. Each
simulation the large ensemble is identical to one others except in their
initial conditions, meaning they differ only in the chaotic fluctuations
internal to the climate system. Thus, this provides one of the largest data
sets of simulations from a single CMIP6-generation ESM from which we can
obtain internal variability to train and test our model on, as it provides a
total of nearly 100,000 months of data. Additionally, we use a set of novel
CESM2 simulations which estimate the climate response to MCB-like
perturbations (described in Appendix 7.2. These are used to evaluate AiBEDO’s
ability to estimate the climate response to a forcing.

####  3.1.1 Spherical Sampling

Earth System Model data is typically stored on 2D latitude-longitude gridded
meshes, which have non-uniform area over the globe with smaller areas at the
poles and larger areas at the equator, complicating their use in ML.
Specifically, the rotational symmetry of the Earth is difficult to represent
using two-dimensional meshes, resulting in inaccurate representations of
significant climate patterns in ML models that assume a two-dimensional format
of data, such as Convolutional Neural Networks (CNNs) or U-Nets. To address
this limitation, we have adopted a geodesy-aware spherical sampling technique
that converts the 2D rectangular grid to a 1D spherical icosahedral mesh,
following the strategy suggested in the recent work Defferrard et al., (2020).
The icosahedral grids consist of equiangular triangles that form a convex
polygon such that the triangles are formed by equally spaced grid points on a
sphere. The number of grid points is defined by their level, with the level-0
grid being an icosahedron, and the number of grid points increasing as
demonstrated in Equation 3:

$N=10\times 2^{2g}+2$ (3)

where $g$ refers to the desired grid level, and $N$ represents the number of
points in the grid that form the icosahedron. This re-sampled icosahedral mesh
provides data with uniform density across the sphere. In this study, we
utilize a level-5 icosahedral grid, equivalent to a 1-D vector of size 10242
with a resolution of approximately 220 $km$. We employ the PyGSP library in
Python to perform the grid transformation. This library is widely utilized for
various graph operations in signal processing and social network analysis,
such as the Erdős-Rényi network Rozemberczki et al., (2020). To convert to
spherical grid, we first develop of a backbone structure for the icosahedral
coordinate system, where the properties of the spherical coordinates (levels),
are specified as inputs. At this stage, the coordinates are represented as
graph networks. Next, latitude and longitude values are assigned to the graph
network points $(x,\leavevmode\nobreak\ y)$ such that they can be expressed in
a geographical coordinate system. Since the points in the icosahedral backbone
do not exactly align with the positions in the 2D gridded Earth System Model
data, we use bilinear interpolation to interpolate the ESM data with the
icosahedral backbone, obtaining the final spherically-sampled data. Figure 2
shows a schematic of grid transformation in the icosahedral spherical sampling
process.

Figure 2: Spherical sampling of ESM data following icosahedral grid.

####  3.1.2 Data-preprocessing

Prior to use by AiBEDO, the CESM2 LE input and output data are preprocessed by
subtracting the ensemble mean for each month and grid point in the data set.
This removes the seasonal cycle of the variables, such that we consider the
month-to-month deviations from the climatology, and removes the secular trend
effects due to external forcings, such that we only consider internal climate
variability. As we are using a LE ensemble mean, we are able to filter out the
forced signal across all time scales, including short-term fluctuations such
as anthropogenic aerosol forcing and volcanic eruptions Rodgers et al.,
(2021).

###  3.2 Data-driven Models of AiBEDO

We develop a function $A_{\tau}$ to map global perturbation of cloud at time t
(input: $\delta f_{t}\in\mathbb{R}^{d\times c_{in}}$ ) to corresponding
climate response with time-lag $l$ (output
$\delta\left<y_{t+l}\right>\in\mathbb{R}^{d\times c_{out}}$). To tackle this,
we formulate the problem as a pixel-wise regression problem, learning a
mapping from input to output, $A^{l}:\mathbb{R}^{d\times
c_{in}}\rightarrow\mathbb{R}^{d\times c_{out}}$. We develop separate models
for different time-lags from 0 (simultaneous) to 6 months at monthly
intervals, and estimate climate response $\delta\left<y\right>$ using a
truncated version of equation 2 with $\tau_{max}=6$ months.

We utilize three machine learning methods to model $A_{\tau}$ which are proven
to be effective for spatio-temporal modeling of ESM data: (1) Spherical
Multilayer Perceptron (S-MLP) Park et al., (2019); Wang et al., (2014), (2)
Spherical U-Net (S-Unet) Mabaso et al., (2006); Ge et al., (2022); Dunham et
al., (2022); Trebing et al., (2021) and (3) Spherical Adaptive Fourier Neural
Operator (S-AFNO) Kurth et al., (2022); Li et al., (2020). Then, we use S-MLP
for our MCB application, since S-MLP performs best out of all three methods.
We describe the three ML methods we employed further below. A schematic of the
three methods is shown in Figure 3. All machine learning models in our work
are trained on a single NVIDIA Tesla V100-SXM2 GPU using 16GB VRAM.

Figure 3: Schematic view of three machine learning methods utilized for AiBEDO
model, $A^{l}$: (a) Multilayer Perceptrons (MLPs), (b) U-Net, (c) Adaptive
Fourier Neural Operator (AFNO).

####  3.2.1 Spherical Multilayer Perceptron (S-MLP)

: MLP is a representative structure of Deep Neural Networks (DNNs) consisting
of input and output layer inter-connected with multiple hidden layers. Each
node in a layer is fully connected with all nodes in previous layer. The
connection between two nodes represents a weighted value that passes through
the connection signal. Each node contains a non-linear activation function to
represent non-linearity of correlation between two connected nodes. The
operation between consecutive layers is defined as multiplication between
nodes in previous layer and corresponding weight parameters, and applying
activation function.

We combine our model architecture with spherical sampling described earlier to
create Spherical Multilayer Perceptron (S-MLP). The S-MLP architecture has
four hidden layers, each containing 1024 nodes. Layer normalization Ba et al.,
(2016) and Gaussian Error Linear Units (Gelu) activation Hendrycks and Gimpel,
(2016) were employed in each layer. The decoupled weight decay regularization
optimization method, AdamW Loshchilov and Hutter, (2017) was utilized to train
our model in an iterative manner. The learning rate was initially set to
$2\times 10^{-4}$ and exponentially decayed at a rate of $1\times 10^{-6}$ per
epoch. We train model for 15 epochs with a batch size of 10. Our S-MLP models
have $\sim$ 108M trainable parameters, and it takes around 1 minute per single
epoch for training on the historical CESM2-LE dataset.

####  3.2.2 Spherical U-Net (S-Unet)

: U-Net is a symmetric U-shaped convolutional neural network for image-to-
image prediction, and consists of a encoder-decoder scheme structure. The
encoder extracts visual features from the input by reducing dimensions in
every layer, and the decoder increases the dimensions and predict output with
same size as input. Encoder and decoder are connected with long skip-
connections allowing high-resolution features from the encoder are combined
with the input of the decoder for better localizing visual feature in
prediction. We chose to explore U-net architecture as they are generally known
to capture fine spatial features in images, and are proven to do well in
biomedical image segmentation Yin et al., (2022), and satellite image analysis
McGlinchy et al., (2019).

We built our S-Unet Autoencoder architecture based on the DeepSphere model
proposed by Defferrard et al. (2020)Defferrard et al., (2020). The decoder and
encoder of the S-Unet were comprised of six Chebyshev Graph Convolutional
LayersBoyaci et al., (2022), followed by spherical Chebyshev pooling, which
performs spherical convolution on 1-D data considering the icosahedral
geometry of the graph structure. The kernel size was set to
[64,128,256,512,512,512] for the encoder and decoder, respectively. The output
of the model was processed through a softmax activation. The S-Unet was
trained using the AdamW optimization method Loshchilov and Hutter, (2017),
with a learning rate of $5\times 10^{-4}$ that was exponentially decayed at a
rate of $1\times 10^{-6}$ per epoch. The model was trained for 30 epochs, and
had approximately 5.8 million trainable parameters. The training of the S-Unet
model took approximately 1.5 minutes per epoch in our computing environment.

####  3.2.3 Spherical Adaptive Fourier Neural Operator (S-AFNO):

Motivated by the recent successes of transformer-based architectures in
climate domain, we adopt the FourcastNetKurth et al., (2022) employing
Adaptive Fourier Neural Operator (AFNO) Guibas et al., (2021) to tackle our
problem. FourcastNet is a weather forecasting model using the multi-layer
transformer architecture employing AFNO inside. The input is first divided
into multiple patches which are embedded in a higher dimensional space with
larger number of latent channels and corresponding positional embeddings.
Then, positional embeddings are formulated as the sequence of tokens. Tokens
are spatially mixed using AFNO followed by subsequent mixing of latent
channels accordingly. Mixed embeddings are passed by MLP to learn higher level
feature. We repeat this process for each transformer layer. After this
process, a linear decoder reconstruct the patches from final embedding.

Instead of the token projection technique employed in ForecastNet, which
involves the composition of tokens from a two-dimensional grid patch of
climatic data, we project all elements of a spherically resampled one-
dimensional input as tokens. These tokens, together with a positional
encoding, are then inputted into a sequence of AFNO layers. Each layer, upon
receiving an input tensor of tokens, performs spatial mixing followed by
channel mixing. We name our model Spherical Adaptive Fourier Neural Operator
(S-AFNO). For S-AFNO model, we used 4-layered transformer Vaswani et al.,
(2017), and set the size of embedding of token as 384. We use GeLu
activationHendrycks and Gimpel, (2016) for the output of MLP layers while
applying layer normalization techniques Ba et al., (2016) to stabilize
training. The S-AFNO model was trained using the AdamW optimization Loshchilov
and Hutter, (2017), with a learning rate of $5\times 10^{-4}$ that was
exponentially decayed at a rate of $1\times 10^{-6}$ per epoch. The model was
trained for 50 epochs, and had approximately 9 million trainable parameters.
The training of the S-AFNO model took approximately 12 minutes per epoch using
CESM2-LE training data in our computing environment.

Figure 4: The performance of AiBEDO in emulating CESM2-LE data using different
ML models including S-MLP, S-Unet, S-AFNO. The persistence curve illustrates
the temporal deviation of the reference output, quantified as the discrepancy
between the ground-truth output at each lag compared with no lag (when lag =
0). Figure 5: Time correlation scores of simultaneous (0-lag) AiBEDO S-MLP
(left column), S-Unet (middle column), and S-AFNO (right column) emulation
versus baseline CESM2 data for surface temperature (top row), precipitation
(middle row), and surface pressure (bottom row). Figure 6: Time correlation
scores of one-month lag AiBEDO S-MLP (left column), S-Unet (middle column),
and S-AFNO (right column) emulation versus baseline CESM2 data for surface
temperature (top row), precipitation (middle row), and surface pressure
(bottom row).

###  3.3 AiBEDO Marine Cloud Brightening Experiments

To evaluate how well AiBEDO estimates climate responses, we compare the impact
of MCB-like forcing in the lag-integrated AiBEDO response to fully coupled
CESM2 simulations Hirasawa et al., (2022). These simulations are summarized in
Table 2 in the Appendix along with a description of how the MCB forcing
perturbations are computed from CESM2 for AiBEDO. Our experiments focus on
perturbing three main regions (shown in Figure 8) for MCB experiments: NEP
(North East Pacific - 0 to 30N and 150W to 110W), SEP (South East Pacific -
30S to 0 and 110W to 70W) and SEA (South East Atlantic - 30S to 0 and 15W to
25E) regions. Following equation 2, for each lag we calculate the AiBEDO
response by taking the mean of the difference between the AiBEDO output for an
$N=480$-month sample of CESM2 internal variability perturbed with the MCB
radiation anomalies Aswathy et al., (2015) minus the AiBEDO output for that
same 480-month sample without perturbations. We use this protocol rather than
simply running AiBEDO with the radiation anomaly fields, because the near-zero
anomalies outside the perturbation regions cause artifacts in the AiBEDO
output, as such a field is entirely unlike any fields the model is trained on.

##  4 Results and Discussion

We evaluate model functionality through two phases of AiBEDO development. The
first phase involves developing an AI emulator of simultaneous and time-lagged
mappings. As described in the previous sections, we developed three AI models
(S-MLP, S-UNet, S-AFNO) and compared their performance in how well they could
emulate the mappings. The second phase encompasses an analysis of the results
from MCB experiments conducted in selected regions on CESM2, along with the
time-integrated outputs generated by AiBEDO through the application of an FDT-
like function. We also compared the time-integrated results across the three
AI models to analyze the performance of different model architectures.

###  4.1 Comparison between AiBEDO ML algorithms (Phase I)

Each machine learning model was trained individually with different monthly
time-lags ranging from 0 to 6 months. The quantitative comparison of emulation
performance of AiBEDO using different ML models are reported in Figure 4. We
evaluate Root Mean Squared Error (RMSE) and spatial correlation score of
prediction with ground truth of our models based on the persistence scores of
the ground truth data. Persistence at a time-lag $t$ is essentially the
temporal deviation of the ground truth outputs at $t$ months from the zero
time-lag instances. The persistence curve at different time-lags is overlaid
with performance curve from our model in Figure 4, which lets us evaluate the
ML models performance over lagged duration. As shown in Figure 4, we can
observe that as the time-lag increases, the predictive accuracy of the model
decreases as anticipated. Notably, the model consistently outperforms
persistence across all time-lags, implying that AiBEDO has learned the
temporal dynamics patterns beyond the simple memory of 0-month temperature
anomalies.

Figures 5 and 6 demonstrate the spatial correlation scores of the emulation of
each ML model (S-MLP, S-Unet and S-AFNO) for simultaneous and one-month time-
lagged model. The corresponding qualitative comparison of sample timesteps is
shown in Figures 10 and 11 in the Appendix. Despite S-AFNO achieving
comparable score to S-MLP in terms of RMSE and spatial correlation for
simultaneous AiBEDO results, as demonstrated in Figures 5 and 10, S-MLP was
found to outperform the other models in capturing global response patterns in
all three variables. The quality deviation of output between the models
becomes evident in the time-lagged response even after a single month (Figure
6), showing higher performance of S-MLP. The qualitative results with a time-
lag demonstrate that S-MLP significantly outperforms the other two models,
S-AFNO and S-Unet, even though RMSE scores are comparable across the models.

The superior performance of S-MLP model can be attributed to its significantly
larger number of trainable parameters and its fully connected network
structure. This allows the model to consider the correlation of each element
in the input, thereby enhancing its ability to regress the results. The fully
connected network structure enables the output to be regressed by multiplying
the different weights of all input elements, which enhances its ability to
capture long-range interactions between locations, potentially driven by the
“butterfly effect”. The subpar performance of the S-Unet can be attributed to
the loss of important information regarding global climate patterns,
particularly related to long-range interactions, during the spherical
convolution and pooling processes. Consequently, machine learning models based
on the assumption of spatial locality, such as the S-Unet, underperform models
that allow for global attention. While S-AFNO performed better than S-Unet,
the model still underperforms compared to S-MLP. The amount of training data
used in this study may not be sufficient for training a transformer-based
S-AFNO model, which typically requires a large amount of data. Moreover, the
time taken to complete one epoch in S-AFNO was already significantly higher
than time per epoch for S-MLP. Adding more parameters and/or data can only
increase the computational requirement for S-AFNO. As a result, S-MLP was
utilized in the subsequent experiments described in this paper.

###  4.2 Response to MCB perturbations (Phase II)

Here, we compare the impact of MCB-like forcing in three regions in the
subtropics on the climate in CESM2 and lag-integrated AiBEDO. Figure 8 shows
the spatial maps of the surface temperature anomalies for the response to each
of the three regions (NEP, SEP, and SEA). The goal of comparing the response
of the CESM2 model and the AiBEDO model to MCB-like perturbations is to
determine if AiBEDO can accurately estimate the forced climate response.
Figure 7 shows that the S-MLP model within AiBEDO has better performance
compared to the other two models, reproducing the climate response pattern
with a correlation score of 0.68 for temperature (tas), 0.51 for precipitation
(pr), and 0.47 for pressure (ps). The correlation score improves with
increased time-integration. However, there are differences in the magnitude of
the responses, which is likely due to missing lags in the integration.

The S-MLP model successfully estimates remote teleconnected responses to MCB
forcing, such as a La Niña-like temperature signal in the Pacific. It also
captures drying in northeast Brazil, central Africa, and southern North
America and Europe, and wetting in the Sahel, south and southeast Asia,
Australia, and central America. The S-Unet and S-AFNO models capture some
aspects of the responses, such as tropical Pacific drying and high pressure
anomalies in the Pacific mid-latitudes. However, they miss many key aspects of
the response patterns. These responses can be used to assess the risk of
regional tipping points, such as Amazon dieback and Sahel region in North
Africa greening McKay et al., (2022) or the reduction in the risk of coral
dieoff due to cooling in the tropical oceans. However, due to the lower
performance of AiBEDO at high latitudes, it may be challenging to evaluate key
cryospheric tipping points such as permafrost loss in Eurasia and North
America.

Figure 7: Comparison of MCB Perturbation Responses in 6-month integrated
S-MLP, S-Unet and S-AFNO Models to CESM2 coupled simulations. Figure 8: CESM2
(left column) and lag-integrated AiBEDO (right column) surface temperature
anomalies due to MCB-like forcing in the NEP (top row), SEP (middle row), and
SEA (bottom row). AiBEDO responses are computed using Simpson’s rule
integration of anomalies from lag 1, 2, 3, 4, 5, 6, 12, 24, 36, and 48 month
models.

The impact of MCB forcing on individual perturbation regions is also evaluated
and compared to CESM2 simulations with similar regional forcing. Figure 8
shows the comparison of perturbation effects on temperature between CESM2 and
S-MLP model of AiBEDO. In general, AiBEDO’s performance is weaker when
considering these regional perturbations compared to when all three regions
are perturbed together. AiBEDO performs best for SEP, followed by SEA and NEP.
The NEP response is overestimated substantially, which may indicate the model
learns too heavily from the El Niño-Southern Oscillation climate pattern
(variations in ocean temperatures in the Pacific Ocean) at the expense of over
modes of variability. However, AiBEDO correctly identifies the climate
responses to the different forcing regions in several key regions. For
example, it correctly identifies that SEP forcing causes La Niña-like cooling
(below-average sea surface temperatures in the central and eastern tropical
Pacific Ocean) and that SEA forcing causes tropical Pacific warming and Amazon
drying. Moreover, AiBEDO performs better in the tropics and over oceans
compared to higher latitudes and over land. This aligns with the regions where
AiBEDO has the highest emulation skill, indicating that the model’s ability to
correctly estimate climate responses to MCB forcing is closely linked to its
ability to emulate internal variability.

###  4.3 Visualization Platform

We developed a front-end interactive visualization platform to facilitate
downstream tasks and exploratory analyses using the suite of trained hybrid AI
models. It allows climate scientists to interact directly with the trained
models (described above) and recreate different MCB experiment scenarios. The
multi-panel design lets the experts load different Earth System Model data,
interactively run the trained hybrid model in the backend and visualize the
model predictions and inputs using popular geospatial projection schemes.
Figure 9 shows an overview of our visualization platform. The leftmost panel
in Figure 9 is the general control panel which permits a subject matter expert
to interact with the models and data source, as well as tune several
experiment parameter for analysis. Figure 9($C_{1}$) highlights the controls
to specify the specific timestamp from the data and the general visualization
projection scheme. Figure 9($C_{2}$) corresponds to the hybrid model controls,
which lets us run the trained AI models with specified data directly from this
interface, as well as clear any previous model prediction results from memory.
Figure 9($C_{3}$) provides the main MCB experiment scenario controls. It helps
to set the important parameters like which geospatial regions to apply MCB
over, which input variable set to perturb and by what extend and running the
perturbed data with the AI models. Figure 9($V_{1}$) and ($V_{2}$) highlight
the input and output visualization panels respectively. Using the set
geospatial projection scheme (as set in $C_{1}$) the different input and
output variables are visualized here. In addition, the output panel has the
options to show the original AI model prediction results, results after MCB,
and the net difference in results as well. This helps to qualitatively analyze
the output predictions and their likely patterns.

Figure 9: High-level overview of our interactive visual analysis system to
work with the trained hybrid models and drive post-hoc analysis for different
MCB scenarios. [source code:https://github.com/subhashis/aibedoviz, demo
video:https://youtu.be/3dmqYqkSLOo]

Since the perturbed input fields prepared as part of the MCB experiments may
lie outside the scope of the input distribution with which the hybrid models
were trained over, it is important to check the out-of-distribution cases. To
address this, in Figure 9($V_{3}$) we visualize the low-dimensional projection
of the MCB perturbed fields over the input distribution to show how far off
the perturbed fields are from the original data. Another important aspect of
our analysis is to understand the impact of MCB on key regional tipping
points. Figure 9($V_{4}$) panel shows the seven regional climate tipping
points that we are tracking with each MCB experiment scenarios. A red color on
this sites indicates that the current MCB setting might trigger factors that
are directly associated with the risk of tipping points in these sites.

##  5 Summary and Future Work

In this study, we introduce a novel methodology for utilizing the Fluctuation-
Dissipation Theorem, a principle derived from statistical mechanics, in the
context of knowledge discovery using AI model to predict the behavior of a
system under external forcing. This approach is particularly useful in
scenario analysis in scientific domains where traditional models are
computationally infeasible. We demonstrate the efficacy of this method through
its application in Marine Cloud Brightening (MCB) climate intervention
analysis, where the number of scenarios required is intractable using
traditional computational models. This constitutes a challenging problem in
climate science, where Earth System Models are employed to simulate changes in
climate and require weeks to complete a single run, and assessing additional
scenarios necessitates additional runs. Our method of training lagged
emulation models of internal variability (noise) and producing time-integrated
outputs using an FDT-like framework presents a viable alternative for fast
prototyping of such scenarios in several scientific domains. In addition to
the central contribution of our work, we also present various components of
AiBEDO that is specific to climate intervention: preprocessing techniques for
large climate datasets, a comparison of machine learning methods for time-
integrated approach, and a user-friendly visualization interface that provides
explainable insights that helps with model investigation and tipping point
analysis, and allows for quick what-if scenario analysis.

Our next steps include expanding the MCB climate intervention framework into
developing optimal pathways for selected tipping point scenarios. Figure 13 in
Appendix shows some of the known tipping points in the climate system and the
associated mitigating strategies that may be used to avoid them. An inverse
search of AiBEDO output may be useful for creating MCB perturbations to
achieve such mitigation measures. For example, to avoid the catastrophic
retreat of the West Antarctic ice sheet, potential MCB perturbation sites
could push the jet circulation equatorward. To achieve this, the user might
query the model to create a forcing scheme to avoid selected tipping points,
and the model would choose optimal spatiotemporal perturbation MCB sites that
could lead to this outcome. We could also add additional constraints not to
tip other climate tipping points in this process. This is a powerful tool that
could help scientists and policymakers understand the climate system’s
teleconnections that could trigger unintended consequences and timely
strategies to avoid catastrophic outcomes due to changing climate.

##  6 Funding and Acknowledgements

The development of AiBEDO is funded under the DARPA AI-assisted Climate
Tipping-point Modeling (ACTM) program under award DARPA-PA-21-04-02. The
authors are grateful for the donation of computing credits and storage from
AWS, which enabled the implementation of MCB scenarios utilizing CESM2 Earth
Systems Model and the training of the AiBEDO model on the cloud.

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##  7 Appendix

###  7.1 Emulation of Climate Noise

AiBEDO emulation performance is evaluated by taking the difference AiBEDO
outputs at a given lag to the corresponding preprocessed CESM2 output field at
the same lag. Example emulation outputs for a single input time step are shown
for the lag-0 (Figure 10) and lag-1 (Figure 11).

Figure 10: AiBEDO emulation results for a sample input time step with no time-
lag (simultaneous run) for surface temperature (tas), precipitation (pr), and
surface pressure (ps) compared with ground truth (first columns) and different
ML models.  Figure 11: AiBEDO emulation results for a sample input time step
with time-lag after one month (lag =1) for surface temperature (tas),
precipitation (pr), and surface pressure (ps) compared with ground truth
(first columns) and different ML models.

###  7.2 MCB Experiment Design

In addition to the CESM2 LE data used to train the model, we use a set of new
CESM2 LE simulations that simulate the impact of MCB on climate Hirasawa et
al., (2022) (Table 2). MCB forcing is imposed by prescribing in-cloud liquid
cloud droplet number concentrations (similar to (Rasch et al.,, 2009), (Jones
et al.,, 2009) and (Stjern et al., 2018a, )) to 600cm-3 in three selected
regions in the northeast Pacific, southeast Pacific, and southeast Atlantic.
We apply the MCB perturbations in the three regions separately and all
together from 2015 to 2065 against a background Shared Socioeconomic Pathway 2
- 4.5Wm-2 (SSP2-4.5) scenario. The CESM2 MCB climate response is thus computed
as the difference between SSP2-4.5 plus MCB simulations minus baseline
SSP2-4.5 (Shared Socioeconomic Pathway 2 - 4.5Wm-2 forcing) simulations.

To compute the radiation perturbations for AiBEDO MCB responses, we use CESM2
“fixed-sea surface temperature” (fixed SST) simulations. In these simulations,
a MCB forcing identical to that used in the coupled simulations is imposed in
the model with SSTs held to climatological values. This allows the computation
of the radiation anomalies in the absence of any radiative feedbacks due to
surface temperature change, which is referred to as the effective radiative
forcing (ERF). Thus, we can compare CESM2 and AiBEDO responses for the same
perturbation fields. AiBEDO perturbation fields are thus calculated as the
annual mean anomalies in cres, crel, cresSurf, crelSurf, netTOAcs, and
netSurfcs anomaly fields (Figure 12) between the Y2000 MCB Perturbed minus
Y2000 Control simulations.

Figure 12: Annual mean radiation anomalies calculated from Fixed SST simulations and applied as MCB perturbations to AiBEDO. Experiment | Role | Forcing | Time span | N
---|---|---|---|---
Historical LE | training, testing, validation | historical | 1850 - 2015 | 50
Y2000 Control | perturbation | Year 2000 Fixed SST | 1 - 20 | N/A
Y2000 MCB Perturbed | perturbation |  | Year 2000 Fixed SST +
---
MCB in NEP, SEP, and SEA
1 - 10 | N/A
SSP2-4.5 LE | response validation | SSP2-4.5 | 2015 - 2100 | 17
SSP2-4.5 + ALL MCB | response validation |  | SSP2-4.5 + 600cm-3 CDNC
---
in NEP, SEP, and SEA
2015 - 2065 | 3
SSP2-4.5 + NEP | response validation | SSP2-4.5 + 600cm-3 CDNC in NEP | 2015 - 2065 | 3
SSP2-4.5 + SEP | response validation | SSP2-4.5 + 600cm-3 CDNC in SEP | 2015 - 2065 | 3
SSP2-4.5 + SEA | response validation | SSP2-4.5 + 600cm-3 CDNC in SEA | 2015 - 2065 | 3
Table 2: CESM2 simulations used to train and verify AiBEDO. NEP, SEP, SEA
denote regions where MCB forcing is imposed, where NEP - Northeast Pacific (0
to 30N; 150W to 110W), SEP - Southeast Pacific (30S to 0; 110W to 70W), SEA -
Southeast Atlantic (0 to 30N; 25W to 15E). Note the fixed SST simulations use
constant climatological conditions, so we do not note specific years for these
simulations. Figure 13: Climate tipping points and mitigation strategies using
AiBEDO framework