PACER: Physics Informed and Uncertainty Aware Climate Emulator

## Abstract

Abstract Physics based numerical climate models serve as critical tools for evaluating the effects of climate change and projecting future climate scenarios. However, the reliance on numerical simulations of physical equations renders them computationally intensive and inefficient. While deep learning methodologies have made significant progress in weather forecasting, they are still unstable for longer roll-out climate emulation task. Here, we propose PACER , a relatively lightweight 2.1M parameter P hysics Informed Uncertainty A ware C limate E mulato R . PACER is trained across is trained across varying spatial resolutions and physics based climate models, enabling faithful and stable emulation of temperature fields at multiple surface levels over a 10 year horizon. We propose an auto-regressive ODE–SDE framework for climate emulation that integrates the fundamental physical law of advection, while being trained under a negative log-likelihood objective to enable principled uncertainty quantification of stochastic variability.
We show PACER’s emulation performance across 20 climate models outperforming relevant baselines and advancing towards explicit physics infusion in ML emulator.

## 1 Introduction

Data-driven weather forecasting models Kochkov et al. (2024); Lam et al. (2023); Nguyen et al. (2023b) have outperformed Numerical Weather Prediction (NWP) (ECMWF, 2023) in recent years for short- to medium-range targets, but their long-lead auto-regressive roll-out becomes numerically unstable over multi-year horizons, undermining reliability for climate modeling (Chattopadhyay & Hassanzadeh, 2023). This necessitates the use of dynamical climate models known as general circulation models (GCMs), which evolve the state forward under specified forcings to yield physically consistent decadal projections.

Physics based dynamical climate models are governed by temporal partial differential equations (PDEs) to describe complex physical processes Gupta & Brandstetter (2022), enabling simulations of climate behavior under various forcing scenarios, such as fluctuating greenhouse gas (GHG) emissions. However, the computational expense associated with solving these PDEs involves the execution of these climate model simulations typically for several months (Balaji et al., 2017).

With the advancement of Artificial Intelligence (AI), data-driven emulators provide a computationally viable substitute. In order to faithfully emulate the dynamical physics based climate model, a Machine Learning (ML) based climate emulator must respect the fundamental physical laws that govern the dynamics of the atmosphere (Watt-Meyer et al., 2023) which improves reliability. Furthermore, accurately capturing the influence of GHG and aerosols is essential for simulating realistic climate responses to different emission scenarios (Bloch-Johnson et al., 2024).

Recently, climate emulators such as ACE (Ai2 Climate Emulator) (Watt-Meyer et al., 2023) and Spherical Dyffusion (Cachay et al., 2024) have shown remarkable and stable results for over 10 year emulation. ACE exhibits physically consistent emulations under prescribed, time-varying sea surface temperatures (SST) and sea-ice boundary conditions, but its deterministic formulation limits the representation of the predictive distribution of climate states or support ensemble generation which aids in better uncertainty quantification. Spherical Dyffusion solves this problem by training on dynamics informed diffusion model known as Dyffusion Cachay et al. (2023) but do not take into account the projected GHG emissions which are essential to the faithful emulation of the climate. Furthermore, both architectures are parameter-heavy, with  200M parameters.

To address these gaps, we propose PACER, a lightweight, physics informed, Neural stochastic differential equation (sde) based, 2.1M parameters climate emulator. PACER integrates GHG emissions data directly into the model’s training framework, to predict climate variables from a given time-series of climate forcer emission maps (GHG and aerosols) allowing for more accurate simulation of future climate states under varying concentration scenarios. Furthermore, we focus on a key phenomenon driving our climate system i.e. advection. In climate modeling, the advection equation is fundamental for simulating the transport of climate variables, such as temperature and moisture (Choi et al., 2023).

Furthermore, PACER is trained with multiple stochastic realizations under direct Negative-log likelihood (NLL) objective which accounts for aleatoric uncertainty and reduces gradient variance. Additionally, modeling the key physical law of advection reduces its dependence on large datasets making it data-efficient and more generalizable across different climate models.
Our contributions are as follows:

- 1.
Physics-guided backbone: We propose PACER with a physics guided backbone and integrate a deterministic advection-forcing PDE by dynamically estimating flow velocities to generate a physically consistent trajectory.
- 2.
Stochastic residual corrector: We refine the ODE trajectory with a stochastic residual corrector. A neural SDE, that learns unresolved/process noise and systematic bias, improving long-horizon fidelity without sacrificing physical structure.
- 3.
Likelihood-based uncertainty: We train the Neural SDE on NLL objective with multiple Brownian realizations per sample $(K>1)$, yielding explicit, heteroscedastic aleatoric uncertainty and reducing gradient variance $\propto 1/K$ for more stable optimization.
- 4.
Finally, we perform extensive experiments across 20 physics based climate models to show PACER is physically consistent and stable for 10 year auto-regressive roll-outs. We also perform zero-shot emulation to highlight generalization capabilities of PACER.

## 2 Related Work

##### Machine Learning (ML) and Physics based Climate Emulators:

Physics based dynamical climate models underpin assessments of future change. These models solve discretized PDEs on a global grid and are computationally prohibitive for multi-decadal integrations, limiting their direct use. Recently, hybrid ML-based and physics-informed climate emulators have been successful in emulating several climate variables at lower wall-clock cost, with strong fidelity across intermediate climates. In literature, climate emulation has been treated as either auto-regressive or diagnostic task. Spherical Fourier Neural Operator (SFNO) based auto-regressive models have shown impressive emulation skills. Cachay et al. (2024) proposed Spherical Dyffusion, a generative emulator of the coarse-grid dataset FV3GFS (see Section [5](https://arxiv.org/html/2410.21657v4#S5.SS0.SSS0.Px3)), which couples dynamics-informed diffusion (DYffusion) with a SFNO.Watt-Meyer et al. (2023) proposed ACE (AI2 Climate Emulator) also based on SFNO for effective physics informed emulation. Another version of ACE (ACE2-ERA5) (Van Loon et al., 2025) is trained on reanalysis data to observe radiative response to changing sea surface temperature patterns. LUCIE and LUCIE3D (Guan et al., 2024; 2025) are also trained on reanalysis data to account for the computational complexity of ACE.

Choi et al. (2023) proposed climate modeling using Graph Neural Networks (GNNs) and Neural ODEs, but do not account for GHG emissions or show any long term stability.
Additionally, there are several climate emulators accounting for emissions and either treat emulation as diagnostic emulation task (Kaltenborn et al., 2023) or single-model emulators that replicate a specific GCM (Scher, 2018; Mansfield et al., 2020; Beusch et al., 2020; Cachay et al., 2021; Watson-Parris et al., 2022; Nguyen et al., 2023a; Kaltenborn et al., 2023).

## 3 Background

##### Neural SDE:

A stochastic differential equation (SDE) models continuous-time evolution with random forcing:

$$ $\mathrm{d}x_{t}=f(x_{t},t)\,\mathrm{d}t+g(x_{t},t)\,\mathrm{d}W_{t}$ (1) $$

where $f$ is the drift, $g$ the diffusion, and $W_{t}$ a Wiener process better known as Brownian motion. The Brownian term injects state/time-dependent stochastic forcing via $g(x_{t},t)$. In Ito form, the state density $p(x,t)$ obeys the forward Kolmogorov (Fokker–Planck) equation describing the time-evolution of the PDE and the related Stratonovich form corresponding the classical chain rule and is coordinate-invariant, which is often advantageous in physics.

Neural SDE introduced by Kidger et al. (2021) replaces $f$ and $g$ with neural networks $f_{\theta}$ and $g_{\theta}$ (Shen & Cheng, 2025). The result is a learned stochastic flow that is continuous in time and probabilistic, which naturally handles irregular sampling, missing data, variable horizons and encodes heteroscedastic aleatoric uncertainty. Training a neural SDE on negative log-likelihood objective with re-parameterized Brownian paths gives calibrated and input-dependent uncertainty.

Wiener (Brownian) process is a continuous path with independent Gaussian increments, so over a step $\Delta t$, the increment satisfies $\Delta W_{t}\sim\mathcal{N}(0,\Delta t)$, where $\mathcal{N}$ is normal (Gaussian) distribution with mean 0 and variance $\Delta t$. In the SDE, $\mathrm{d}W_{t}$ provides white-noise forcing representing fast, unresolved processes. In discrete time the increment is sampled as $\Delta W_{t}\approx\sqrt{\Delta t}\,\varepsilon$ with $\varepsilon\sim\mathcal{N}(0,I)$ with standard multivariate normal and identity covariance $I$. In equation [1](https://arxiv.org/html/2410.21657v4#S3.E1), the drift network $f_{\theta}$ sets the mean tendency, while the diffusion $g_{\phi}(x,t)$ scales this stochastic input state-dependently.

Sampling different Brownian sequences $\{\varepsilon_{k}\}$ yields different trajectories from the same initial condition, forming an ensemble, aggregating many such realizations characterizes the predictive distribution and when used in NLL training, reduces gradient variance by averaging over noise paths.

## 4 Methodology

### 4.1 Problem Statement

We model climate emulation as a probabilistic continuous-time, auto-regressive task $P_{\theta}\!\left(x_{t_{1}:t_{H}}\,\middle|\,x_{t_{0}},\,f_{[t_{0},t_{H}]}\right)$. The goal is to learn the climate variable trajectory $(x_{t_{1}:t_{H}})$ over a time horizon $H$ given an initial state $x_{t_{0}}=x_{0}\in\mathbb{R}^{C\times X\times Y}$, where $C$ is the 2D climate variable of interest driven by time-varying GHG forcings $F$, with $f_{t}\in\mathbb{R}^{F\times X\times Y}$.

The dynamics are learned over two stages.

Physics backbone ODE: We first integrate an advection-forcing partial differential equation (PDE) to obtain a deterministic physics trajectory, considering that climate system evolves according to a 2D advection-forcing process.

Neural SDE refinement: We then refine the ODE physics trajectory with a neural SDE residual driven by multiple Brownian paths to capture unresolved variability.

Formally, let $x_{t}\in\mathbb{R}^{C\times X\times Y}$ be the climate state at time $t$ on $X\times Y$ latitude-longitude spatial grid and $F_{t:t+\Delta t}\in\mathbb{R}^{F\times X\times Y}$ be prescribed GHG forcings over $[t+\Delta t]$. We learn the climate trajectory as

$$ $x_{t+\Delta t}=\mathcal{M}_{\theta}\!\big(x_{t},F_{t:\,t+\Delta t}\big)=\underbrace{\Phi_{\Delta t}\!\big(x_{t},F_{t:\,t+\Delta t}\big)}_{\text{ODE: advection + forcings}}\;+\;\underbrace{R_{\theta}\!\big(x_{t},F_{t:\,t+\Delta t}\big)}_{\text{neural residual refinement}}$ (2) $$

where $\mathcal{M}_{\theta}$ is the learned one-step update operator that maps the current state and forcings to the next state. $\Phi_{\Delta t}$ is the ODE solver and $R_{\theta}$ is the neural residual module parameterized by $\theta$, which refines the solution to the advection-forcing equation given the input emissions $F$. Iterating this step auto-regressively from $x_{0}$ under $F_{0:H}$ yields the trajectory to the time horizon $H$. The model is thus designed to learn the spatiotemporal patterns of our climate system dictated by the underlying physical processes modeled by the PDE.

### 4.2 Advection Process

We model climate emulation as a continuous spatio-temporal process that captures the fundamental physical process of advection under GHG forcings, which together dictate how substances are transported throughout the climate system. The general form of the advection-forcing equation in a climate system is defined in equation [3](https://arxiv.org/html/2410.21657v4#S4.E3).

$$ $\frac{\partial u}{\partial t}\;+\;v_{x}\,\frac{\partial u}{\partial x}\;+\;v_{y}\,\frac{\partial u}{\partial y}\;=\;S\!\left(u,\,F;\,x,y,t\right),$ (3) $$

where $u$ is the physical variable under consideration and $S$ represents the ghg forces. To faithfully emulate the climate’s chaotic nature, it is essential to determine the path and rate at which the physical quantities are transported given by $v\cdot\nabla u$ where $v$ is the velocity vector of the fluid (e.g., wind velocity) and $\nabla u$ is the gradient of the quantity being transported (e.g., temperature).

We use an ODE solver to solve the 2D advection-forcing equation [3](https://arxiv.org/html/2410.21657v4#S4.E3) by discretizing the spatial domain using the centered Finite Difference Method (FDM) Fiadeiro & Veronis (1977) considering the earth is divided into spatial grid points in x and y directions (longitude x latitude). FDM employ spatial discretization to approximate derivatives using the values at grid points (Molenkamp, 1968; LeVeque, 2007).
We explain the spatial discretization and show it’s effect visually in Appendix [B.3](https://arxiv.org/html/2410.21657v4#A2.SS3). The spatial derivatives are therefore descritized as equation [4](https://arxiv.org/html/2410.21657v4#S4.E4)

$$ $\frac{\partial u}{\partial x}(x_{i},y_{j},t^{n})\approx\frac{u_{i+1,j}^{n}-u_{i-1,j}^{n}}{2\,\Delta x},\qquad\frac{\partial u}{\partial y}(x_{i},y_{j},t^{n})\approx\frac{u_{i,j+1}^{n}-u_{i,j-1}^{n}}{2\,\Delta y}.$ (4) $$

The resulting semi-discrete system (continuous in $t$) is integrated with a Dormand–Prince ODE solver (dopri5) (Dormand & Prince, 1980).

### 4.3 Velocity Inference

Quantifying the temporal change in velocity of climate variables over the sampling grid is essential for accurate climate modeling. Because global and regional models evolve climate fields on a fixed grid, their temporal trends must be translated into spatial displacements. The standard approach is the climate-velocity index (Gaponenko et al., 2022). For a state variable $u$ (e.g., temperature), the local migration speed is given by equation [5](https://arxiv.org/html/2410.21657v4#S4.E5)

$$ $v(x)\;=\;\left\lVert\nabla u\right\rVert\,\frac{\partial u}{\partial t}$ (5) $$

Here, we infer a vector transport field $\mathbf{v}(x,y,t)$ by inverting an advection balance for the predicted state variables. We empirically estimate the initial velocity from data using temporal derivative from a natural cubic spline fit McKinley & Levine (1998), and spatial gradients via finite differences, then improve the initialization with a small residual multi-layer perceptron during ODE integration to obtain a directional flow field used prognostically in the advection–forcing ODE.

##### Velocity initialization from past states.

Let $u_{\text{past}}\in\mathbb{R}^{B\times T\times C\times\text{lat}\times\text{lon}}$ be a short history (where $T=timesteps$) of $C$ *state/target channels* on a longitude-latitude grid. For each channel $i\in\{1,\dots,C\}$ and grid point $(x,y)$, fit a natural cubic spline $s_{i}(t;x,y)$ to $\{u_{\text{past}}(\cdot,t_{k},i,x,y)\}_{k=0}^{T-1}$ and extract

$$ $\dot{u}_{i}(x,y)\;=\;\partial_{t}s_{i}(t;x,y)\big|_{t=t_{0}}.$ $$

We compute centered finite‐difference spatial gradients at $t_{0}$,

$$ $\partial_{x}u_{i}(x,y),\qquad\partial_{y}u_{i}(x,y),$ $$

and a stabilizing Laplacian

$$ $\Delta u_{i}(x,y)\;=\;\partial_{xx}u_{i}(x,y)\;+\;\partial_{yy}u_{i}(x,y).$ $$

We set $\alpha=10^{-7}$; this is a small stabilization constant to avoid division by near-zero $\partial_{x}u,\partial_{y}u.$ as shown in equation equation [6](https://arxiv.org/html/2410.21657v4#S4.E6)

$$ $v_{x}\;\mathrel{+}=\;\frac{-\,\dot{u}_{i}\;-\;u_{i}\,\Delta u_{i}}{\partial_{x}u_{i}+\alpha},\qquad v_{y}\;\mathrel{+}=\;\frac{-\,\dot{u}_{i}\;-\;u_{i}\,\Delta u_{i}}{\partial_{y}u_{i}+\alpha}$ (6) $$

##### Velocity refinement:

Given the current state $u(t)\in\mathbb{R}^{B\times C\times\text{lat}\times\text{lon}}$ and previous velocity $v(t)\in\mathbb{R}^{B\times C\times\text{lat}\times\text{lon}}$, we implement a simple 2-layer $1{\times}1$ convolutional MLP which produces

$$ $\Delta v(u)\;=\;\underbrace{W_{2}^{(1\times 1)}}_{\;32\to 2}\;\sigma\!\left(\underbrace{W_{1}^{(1\times 1)}\!\ast u}_{\;2\to 32}\right).$ (7) $$

where $*$ is a convolution operator, $\sigma$ represents a ReLU non-linearity, $W_{n}$ terms are the layer $n$ weights, and applies a residual update

$$ $v_{\mathrm{ref}}(t)\;=\;v(t)\;+\;\Delta v\!\big(u(t)\big).$ $$

The advection–forcing dynamics use $v_{\mathrm{ref}}$,

$$ $\partial_{t}u\;=\;-\,v_{\mathrm{ref},x}\,\partial_{x}u\;-\;v_{\mathrm{ref},y}\,\partial_{y}u\;+\;S(u,F),$ $$

and the velocity relaxes toward the refined field,

$$ $\partial_{t}v\;=\;v_{\mathrm{ref}}\;-\;v,$ $$

The refined velocity is then used in the original pde solved by dopri5 solver in equation [3](https://arxiv.org/html/2410.21657v4#S4.E3). The refiner learns a state-dependent residual and learns transport corrections directly from loss signals on $u$, improving downstream forecast error without breaking the physics structure.

### 4.4 Neural SDE for Trajectory Refinement

The physics-based trajectory produced by the ODE is then refined using a unet based Neural SDE. Given an ODE–advanced state sequence $x_{\text{next}}(t)$ over a short window, we encode it to a latent initial condition given in equation [8](https://arxiv.org/html/2410.21657v4#S4.E8)

$$ $z_{0}\;=\;\mathcal{E}\!\big(x_{\text{next}}\big).$ (8) $$

where $\mathcal{E}$ is a 3D encoder that maps the physical fields in the current window to a compact latent state $z_{0}$ that summarizes the flow‐relevant information for stochastic refinement. We then evolve $z(t)$ by an Itô SDE with *diagonal* noise given by equation [9](https://arxiv.org/html/2410.21657v4#S4.E9)

$$ $\mathrm{d}z_{t}\;=\;f_{\theta}(z_{t},t)\,\mathrm{d}t\;+\;g_{\theta}(z_{t},t)\,\mathrm{d}W_{t},\qquad z_{t_{0}}=z_{0},$ (9) $$

where $W_{t}$ is a $d$–dimensional Wiener process, $f_{\theta}$ (drift) and $g_{\theta}$ (diffusion) are convolutional maps defined on the latent grid. We parameterize the SDE coefficients as *convolutional operators* on a latent grid. Let
$\mathcal{Z}$ denote the latent field space (channels$\times$height$\times$width). We learn

$$ $f_{\theta}:\ \mathcal{Z}\rightarrow\mathcal{Z},\qquad g_{\theta}:\ \mathcal{Z}\rightarrow\mathcal{Z},$ (10) $$

where $f_{\theta}$ (drift) is implemented as a *multiscale unet style encoder-decoder with skip connections* followed by a $1{\times}1$ convolution to produce a residual tendency, and $g_{\theta}$ (diffusion) is a shallow 2D convolutional encoder–decoder that maps the latent state $z_{t}$ to a per-pixel, per-channel diffusion amplitude for the Neural SDE. This construction preserves spatial locality and translation equivariance, and yields heteroscedastic uncertainty via $g_{\theta}$. We integrate with a stochastic runge-kutta (srk) as shown in equation [11](https://arxiv.org/html/2410.21657v4#S4.E11)

$$ $z_{t_{1}}\;=\;\texttt{sdeint}\big(f_{\theta},g_{\theta},\,z_{0},\,\{t_{k}\}_{k=0}^{K}\big),$ (11) $$

using a *Brownian interval* $W_{t}$ over $[t_{0},t_{K}]$. The refined
physical-space trajectory over the window is obtained by decoding the output of neural sde back to physical space as given in equation [12](https://arxiv.org/html/2410.21657v4#S4.E12)

$$ $\hat{x}(t_{1}{:}t_{K})\;=\;\mathcal{D}\!\big(z_{t_{1}{:}t_{K}},\,x_{\text{skip}}\big).$ (12) $$

where $\mathcal{D}$ is a 3D decoder (transpose conv + concatenation with $x_{skip}$) which reconstructs refined fields over the window; the skip tensor $x_{\text{skip}}$ injects mid-level encoder features to preserve spatial detail and stabilize reconstruction. The complete architecture and encoder-decoder structure is shown in Figure [1](https://arxiv.org/html/2410.21657v4#S4.F1).

##### Ensemble refinement and uncertainty.

To quantify process uncertainty, we draw $K$ independent Brownian paths and solve the SDE for each:

$$ $z^{(k)}\;=\;\text{SDE_integrate}\big(f_{\theta},g_{\theta},\,z_{0},\,\{t_{j}\};\,W^{(k)}\big),\qquad\hat{x}^{(k)}\;=\;\mathcal{D}\!\big(z^{(k)},x_{\text{skip}}\big),\quad k=1,\dots,K.$ (13) $$

Each realization uses the same initial condition $z_{0}$ but a different Brownian path $W^{(k)}$, yielding a
sample from the predictive distribution over trajectories in the window.

We summarize the ensemble with the mean and variance:

$$ $\mu\;=\;\frac{1}{K}\sum_{k=1}^{K}\hat{x}^{(k)},\qquad\Sigma\;=\;\frac{1}{K-1}\sum_{k=1}^{K}\big(\hat{x}^{(k)}-\mu\big)^{\!2}.$ (14) $$

The mean $\mu$ serves as the refined state passed to the next auto-regressive window, while $\Sigma$ provides a time–space–channel uncertainty estimate suitable for calibration and downstream risk
assessment.

### 4.5 Uncertainty Quantification

The existing climate emulators including the probabilistic ones such as spherical Dyffusion by Cachay et al. (2024) are trained with a *deterministic* objective such as MSE/RMSE, which fits the conditional mean but leaves the predictive spread under-determined. To learn both the mean and a heteroscedastic variance, we train with a latitude-weighted Gaussian negative log–likelihood (NLL) objective.

Consider a global dataset $\mathcal{D}=\{(t_{i},y_{i})\}_{i=1}^{N}$, where each frame $y_{i}\in\mathbb{R}^{C\times H\times W}$ lies on a regular lat–lon grid. At time $t_{i}$ the model predicts cell-wise mean $\mu_{\theta}(t_{i})$ and variance $\sigma^{2}_{\theta}(t_{i})$. The latitude-weighted Gaussian negative log-likelihood with a variance prior is given by equation [15](https://arxiv.org/html/2410.21657v4#S4.E15)

$$ $\mathcal{NLL}=-\frac{1}{NCHW}\sum_{i=1}^{N}\Bigg[\sum_{p=1}^{CHW}\tilde{w}_{p}\;\log\mathcal{N}\!\big(y_{i,p}\,\big|\,\mu_{\theta,i,p},\,\sigma^{2}_{\theta,i,p}\big)\;+\;\log\mathcal{N}^{+}\!\big(\sigma^{2}_{\theta,i}\,\big|\,0,\,\lambda_{\sigma}^{2}\mathbf{I}\big)\Bigg]$ (15) $$

where $\mathcal{D}=\{(t_{i},y_{i})\}_{i=1}^{N}$: dataset of $N$ time points; each frame $y_{i}\in\mathbb{R}^{C\times H\times W}$. Index $p=1,\dots,CHW$ is a flattened index over channel $c$, latitude row $\ell$, longitude column $m$; $y_{i,p}$, $\mu_{\theta,i,p}$, and $\sigma^{2}_{\theta,i,p}$ denote the observed value, predicted mean,
and predicted variance at a single space–channel location of frame $i$; $\tilde{w}_{p}$ is the latitude weight for location $p$,
$\log\mathcal{N}$ is Gaussian log-likelihood at location $p$;
$\log\mathcal{N}^{+}$ is half-Gaussian prior over the
non-negative variance vector of frame $i$ with scale $\lambda_{\sigma}$ which shrinks extreme variances and stabilizes training.

Figure: Figure 1: Complete architectural pipeline of PACER.
Refer to caption: x1.png

## 5 Experiments and Results

##### Task:

The goal of PACER is to auto-regressively emulate state variables from initial condition and a parallel time series of ghg emissions. We train our model on 1-year and validate on 10-year roll-outs. We also validate the generalisation of our methodology using zero-shot emulation. We compare PACER against Unet and ConvLSTM baselines provided by ClimateSet. We also compare against a simple SFNO Bonev et al. (2023) given in LUCIE. The hyperparameter details for adaptation of all baselines are given in Appendix [C](https://arxiv.org/html/2410.21657v4#A3).

##### ClimateSet:

We train PACER on a total of 20 climate models, 19 provided by ClimateSet Kaltenborn et al. (2023) and FV3GFS (Zhou et al., 2019). ClimateSet compiles climate data from the Coupled Model Intercomparison Project Phase 6 (CMIP6) (Eyring et al., 2016) , incorporating climate model outputs from ScenarioMIP (O’Neill et al., 2016) and future emission trajectories of climate forcing agents from Input Datasets for Model Intercomparison Projects (Input4MIPs) (Durack et al., 2017).
Each CMIP6 climate model has been standardized to a spatial resolution of 250km i.e. 96 $\times$ 144 grid points (latitude $\times$ longitude) with a monthly temporal resolution. Both input and output datasets consist of 86-year time-series data spanning four SSP scenarios (SSP1-2.6, SSP2-4.5, SSP3-7.0, SSP5-8.5) from 2015 to 2100. We train on SSP1-2.6, validate on SSP3-7.0, and test on SSP2-4.5 scenarios.

##### FV3GFS:

We also train PACER on FV3GFS, a climate model used at the US National Weather Service and US National Centers for Environmental Prediction. It consists of 11-member initial-condition ensemble, each a 10-year integration saved every 6 hours. Forcings consist of annually repeating climatological sea-surface temperature (1982–2012 mean) and top-of-atmosphere insolation. Model output is conservatively regridded from the FV3 cubed-sphere to a $1^{\circ}$ Gaussian grid $180\times 360$ grid point and passed through a spherical-harmonic analysis–synthesis to suppress high-latitude artifacts. We train on 100 years drawn from 10 ensemble members and evaluate on a distinct 10-year member. We downsample it to a monthly cadence to align with ClimateSet baselines and ensure consistent training.
The detailed list of input (diagnostic and prognostic) and output variables (diagnostic) is given in Appendix [D](https://arxiv.org/html/2410.21657v4#A4)

##### Evaluation Metrics

We evaluate all models using latitude-weighted Root Mean Square Error (RMSE) for deterministic approach Nguyen et al. (2023a) and Continuous Rank Probability Square (CRPS) Winkler et al. (1996) for probabilistic approach which are described in Appendix [B.1](https://arxiv.org/html/2410.21657v4#A2.SS1).

### 5.1 Results

We report RMSE and CRPS for PACER, UNet, ConvLSTM and SFNO. The results for all ClimateSet’s physics based numerical climate models are shown in Table [1](https://arxiv.org/html/2410.21657v4#S5.T1). The results for emulating temperature at different surface levels for physics based climate model FV3GFS are given in Table [2](https://arxiv.org/html/2410.21657v4#S5.T2). The best performing ML based climate emulators are emboldened. PACER outperforms all ML based models for emulating surface air temperature for auto-regressive 10 year long simulations across 20 climate models.

### 5.2 Zero-shot Emulation

We perform zero-shot emulation experiments to test the generalization capabilities of ML models on three physics based climate models: AWI-CM-1-1-MR, EC-Earth3 and TaiESM1. We report the RMSE and CRPS for TAS (surface air temperature) pre-trained on different climate models and tested on these three climate models and results are shown in Table [3](https://arxiv.org/html/2410.21657v4#S5.T3). The metric for best performing model is emboldened. The pretrained climate model column shows which dataset the ML model was initially trained on before being tested on the either of the three climate models. Overall, PACER outperforms all ML models on 1-year auto-regressive roll-out for zero-shot emulation.

Table: Table 1: 10-year roll-out results on 19 climate models which are a subset of ClimateSet. We report the RMSE and CRPS for TAS (surface air temperature).

Table: Table 2: 10-year auto-regressive roll-outs on the physics based on physics based FV3GFS dataset. We evaluate near-surface air temperature (upto seven vertical levels) using RMSE and CRPS aggregated over space and time across the roll-out horizon. (Lower is better).

Figure: Figure 2: Time–mean maps for air temperature fields across vertical levels (2,3,4,5). Both the quantitative results in table [2](https://arxiv.org/html/2410.21657v4#S5.T2) and visual results indicate stronger fidelity at higher levels (L4–L5) than at lower tropospheric levels (L2–L3), with reduced discretizations.
Refer to caption: air_temperature_2_true.png

**Table 3: Zero-shot Emulation results for 1-year roll-out on AWI-CM-1-1-MR, EC-Earth3 and TaiESM1**
| Climate Model | PACER | UNet | ConvLSTM | SFNO |  |
| --- | --- | --- | --- | --- | --- |
| RMSE | CRPS | RMSE | RMSE | RMSE |  |
| AWI-CM-1-1-MR | 0.190 | 0.134 | 0.534 | 0.589 | 0.322 |
| EC-Earth3 | 0.214 | 0.149 | 0.759 | 0.590 | 0.342 |
| GFDL-ESM4 | 0.198 | 0.136 | 0.603 | 0.599 | 0.325 |
| FGOALS-f3-L | 0.225 | 0.155 | 0.659 | 0.587 | 0.351 |

### 5.3 Ablation Studies

##### Physics informed vs Physics Uninformed:

Figure: Figure 3: Physics vs. no-physics across four CMIP6 climate models
Refer to caption: x2.png

We perform an ablation study in which we remove the advection-forcing pde and directly train on the neural sde. We observe a decline in the performance of emulation and show it in figure [3](https://arxiv.org/html/2410.21657v4#S5.F3). The study shows that although climate evolves under coupled, multi-physics PDEs (Navier–Stokes, thermodynamics, etc.), injecting even a minimal physics informed advection with prescribed forcings, improves skill as compared to purely data-driven baseline.

## 6 Conclusion and Future Work

In this work, we present PACER, a physics and uncertainty aware climate emulator which accounts for Earth’s atmospheric advection phenomenon. We incorporate a key physical law in PACER by solving a time-dependent partial different equation (PDE) using an ODE solver. Additionally, we refine the inconsistencies of the generated trajectory through a Neural SDE and account for uncertainties by explicitly training on a negative log-liklihood objective.

However, there are a few potential limitations of PACER which will be addressed as a possible future work. Our model only accounts for Advection physical law, however beyond advection, climate models solve the rotating (hydrostatic/primitive) Navier–Stokes system with moist thermodynamics, radiative transfer, turbulence/mixing and cloud microphysics, coupled to ocean circulation, land surface, and sea-ice to simulate a full multi-physics Earth system. We consider this work as a step towards building fully physics compatible climate emulators.

Additionally, PACER is trained on coarse resolution data which does not fully account for extreme events at regional level. These limitations can be addressed by training on high resolution data and encoding complex physical constraints to be able to fully emulate physics based climate models in a reliable and compute efficient way.

#### Acknowledgments

This work was supported by SmartSAT Cooperative Research Cen-
ter(project number P3-31s). We would also like to acknowledge
National Computational Infrastructure (NCI), a high performance
computing center for providing us with GPU resources and data
collection which enabled us to perform this research.

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## Appendix A Impact Statement

Our research presents a deep learning based climate emulator which emulates temperature for multiple climate models by solving atmospheric advection. ML climate emulators run much faster and use far less energy than Global Climate Models (GCMs), making multi-year simulations with thousands of runs, quick and practical to do routinely. While ML-based climate emulators have advanced rapidly, they still lag state-of-the-art Earth System Models (ESMs) in process fidelity, long-horizon stability, and the representation of extremes. This work moves one step closer to parity by coupling a physics-aware backbone with a probabilistic correction mechanism, delivering stable 10 year roll-outs. Looking ahead, replacing full-physics solves with learned surrogates can cut runtime and energy use by orders of magnitude while preserving target statistics (means, variability, and extremes), making large ensembles, and uncertainty quantification fast and reliable, thereby accelerating climate risk assessment and reducing the carbon footprint of climate computation.

## Appendix B PACER Training Details

### B.1 Evaluation Metrics

The latitude-weighted Root Mean Square Error (RMSE) is given in equation [16](https://arxiv.org/html/2410.21657v4#A2.E16) and Continuous Rank Probability Square (CRPS) for probabilistic approach is given in equation [17](https://arxiv.org/html/2410.21657v4#A2.E17).

$$ $RMSE=\frac{1}{N}\sum_{t}^{N}\sqrt{\frac{1}{HW}\sum_{h}^{H}\sum_{w}^{W}L(i)(y_{thw}-pred_{thw})^{2}}$ (16) $$

where $L(i)$ accounts for latitude weights.
$L(i)=\cos(lat(i))/\frac{1}{H}\sideset{}{{}_{i^{\prime}=1}^{H}}{\sum}\cos(lat(i^{\prime}))$.

$$ $\mathrm{CRPS}\big(\mathcal{N}(\mu,\sigma^{2}),\,y\big)=\sigma\!\left[\,z\big(2\Phi(z)-1\big)+2\phi(z)-\frac{1}{\sqrt{\pi}}\right],\qquad z=\frac{y-\mu}{\sigma},$ (17) $$

### B.2 Hyperparameters

**Table 4: Left: PACER Training Details and Right: PACER ConvLayer Details**
| Hyperparameters | Value |
| --- | --- |
| Epochs | 30 |
| Conv2d kernel size | 3x3 |
| Integrator | srk |
| Noise type | diagnol |
| ODE solver | dopri5 |
| Pooling No.1 | AvgPool2d (2,2) |
| Normalization | GroupNorm |
| Activation Function | ReLU |
| Optimizer | Adam |
| Learning Rate | $1e-4$ |
| Weight Decay | $1e-6$ |
| Adam $\epsilon$ | $1e-8$ |
| LR scheduler | Exponential decay |
| Scheduler gamma | $0.98$ |
| Batch size | 4 |

### B.3 Numerical Discretization: Impact of Finite Difference Methods (FDM) on Earth’s Spatial Gridding in Climate Models

We utilize FDM for spatial discretization in PACER which divides the the physical space into a grid of discrete points. Each grid point represents a specific location, and the value of the physical quantity (e.g., temperature) is computed at each point. In FDM, continuous differential equations are approximated using discrete differences between values at specific grid points. The process of discretization converts the continuous space into a finite grid, and the differential operators (like derivatives) are approximated using differences between the values at neighboring grid points.

Figure: Figure 4: Numerical discretization effect on emulation of Temperature.
Refer to caption: x3.png

### B.4 Hardware and Software Requirements

We use PyTorch Paszke et al. (2019), Pytorch Lightning Falcon (2019), torchdiffeq Chen et al. (2018) for implementation of PACER. We perform all emulator training experiments on a single NVIDIA H100_NVL GPU.

## Appendix C Baseline Hyperparameters

We follow the same training setting for both Unet and ConvLSTM as described in ClimateSet Kaltenborn et al. (2023) except changing it from diagnostic to auto-regressive settings. We maintain the hyperparameters of the SFNO consistent with the configuration proposed in LUCIE (Guan et al., 2024).

**Table 5: UNET Training Details**
| Hyperparameters | Value |
| --- | --- |
| Encoder Backbone | VGG11 pre-trained on ImageNet |
| Library | segmentation-models-pytorch (SMP) |
| Encoder stride constraint | 32 (downsampling factor) |
| Input grid handling | Zero-pad lon/lat to nearest multiple of 32 |
| Output resizing | Adapted average pooling to original grid size |
| Decoder | Standard U-Net decoder (upsampling + skip connections) |
| Readout head | Linear layer |
| Temporal wrapper | Time-Distributed layer around U-Net |
| Optimizer | Adam |
| Learning Rate | $2e-4$ |
| Weight Decay | $1e-6$ |
| Adam $\epsilon$ | $1e-8$ |
| LR scheduler | Exponential decay |
| Scheduler gamma | $0.98$ |

**Table 6: Left: ConvLSTM Training Details and Right: SFNO Training Details**
| Hyperparameters | Value |
| --- | --- |
| Conv2d filters | 20 |
| Conv2d kernel size | 3x3 |
| Activation Function | ReLU |
| Pooling No.1 | AvgPool2d (2,2) |
| Pooling No.2 | AvgPool2d (lon/2, lat/2) |
| LSTM Layers | 1 |
| LSTM hidden units | 25 |
| Optimizer | Adam |
| Learning Rate | $2e-4$ |
| Weight Decay | $1e-6$ |
| Adam $\epsilon$ | $1e-8$ |
| LR scheduler | Exponential decay |
| Scheduler gamma | $0.98$ |

## Appendix D Dataset Variables

**Table 7: ClimateSet Variables**
| Symbol | Description | Usage |
| --- | --- | --- |
| TAS | Surface Air Temperature | Prognostic Variable (input & output) |
| PR | Precipitation | Prognostic Variable (input & output) |
| $CO_{2}$ | Carbon Dioxide | Forcing Variable (input only) |
| $CH_{4}$ | Methane | Forcing Variable (input only) |
| $SO_{2}$ | Sulfur Dioxide | Forcing Variable (input only) |
| BC | Black Carbon | Forcing Variable (input only) |

**Table 8: FV3GFS Variables. The k subscript refers to a vertical layer index and ranges from 0 to 7**
| Symbol | Description | Usage |
| --- | --- | --- |
| $T_{k}$ | Air Temperature | Prognostic Variable (input & output) |
| DSWRFsfc | Downward shortwave radiative flux at surface | Forcing Variable (input only) |
| DLWRFsfc | Downward longwave radiative flux at surface | Forcing Variable (input only) |
| ULWRFsfc | Upward longwave radiative flux at surface | Forcing Variable (input only) |
| USWRFsfc | Upward shortwave radiative flux at surface | Forcing Variable (input only) |

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 2
presentation: 1
questions: See above.
soundness: 2
strengths: The method beats the baselines, and is able to learn climate.
summary: The paper proposes physics-aware architecture for climate. The method is based on advection, and incorporates a neural SDE.
weaknesses: The clarity of the paper is subpar. There are typos, the math is inconsistent and the method pipeline is difficult to follow. The model is given as a long storyline describing different parts of the system, without giving a clear beginning or overall framework. I'm quite confused how the SDE, the ODE and the latent modelling plays together, or what is being learnt.  

The novelty is low: the paper combines relatively straightforward PDE equations with a standard neural SDE. 

The baselines are overly simple. Comparing to just Unet or LSTM is ok as naive baselines, but they are not serious competing methods. The SFNO is a good baseline, but there is a total lack of any bespoke climate or physics models for this problem. The results lack standard deviations.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 1
questions: 1. Can you provide quantitative evidence that learned velocity fields (after refinement) maintain physical plausibility? For example, the autghors can show visualizations compared to reanalysis winds.
2. Can you compare performance using Euler and RK4 solvers instead of the dopri5 solver? Can you compare RMSE and wall-clock time for different ODE solvers?
3. Do PACER and the baselines have the same number of encoder and decoder layers? What are the parameter counts for PACER and each baseline? Could you provide comparisons under a reasonably fair parameter budget?
4. Why was no direct comparison performed with studies [1,2] that similarly integrated the Advection-Diffusion Equation into a Neural Network framework? Can you discuss further what differentiates your work from these related studies?


> [1] "Climate modeling with neural advection–diffusion equation." Knowledge and Information Systems 65.6 (2023): 2403-2427.
> 
> [2] “Climode: Climate and weather forecasting with physics-informed neural odes." ICLR 2024
soundness: 2
strengths: 1. The paper proposes a novel autoregressive framework that combines an advection PDE-based physics-informed backbone with a Neural SDE-based uncertainty-aware residual corrector.
2. Training the neural SDE under a Gaussian NLL objective with multiple Brownian realizations ($K>1$) enables heteroscedastic aleatoric uncertainty estimation and reduces gradient variance.
3. The explicit integration of advection dynamics via ODE solver (dopri5) with velocity field inference provides a transparent physics backbone that is computationally lightweight.
summary: PACER is an uncertainty-aware climate emulator that integrates a physics-based advection PDE with a Neural SDE residual corrector. The proposed method enables stable 10-year temperature field emulation on multiple climate models.
weaknesses: 1. Despite claiming to be "Physics Informed," the physical laws integrated into the emulator are limited only to advection, which is far from the complex multi-physics systems of actual GCMs.
2. PACER emphasizes being "lightweight" (2.1M parameters) compared to ACE (200M parameters) and Spherical Dyffusion (200M parameters), but in the results section, it only compares performance with UNet, ConvLSTM, and SFNO instead of these models.
3. There is only one ablation (Figure 3) that compares "physics informed vs uninformed" but doesn't isolate individual components.
4. The notation lacks consistency, reducing readability. Eq.2 uses $x_t$, while subsequent sections use $u_t$ for the climate state. Forcing is denoted as $F$ in some places and $f$ in others.
5. The authors claim their proposed method is "lightweight," but there is no wall-clock time comparison, GPU memory usage, or FLOP analysis.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 1
questions: 1. Can you clarify the velocity inference method outlined in Section 4.3? Why is this necessary at all? What exactly is $v$ in Eq. 5? Did you use the equation exactly as it appears in Eq. 5, with the wrong units? If so, how does this affect your results?
2. Why is an advection-based prior an appropriate choice for TAS and precipitation?
3. Can you provide wall-clock timing comparisons against GCMs and baseline emulators?
4. How exactly is $S(u, F; x, y, t)$ parameterized? How do GHG forcings enter the PDE?
5. How are spherical geometry boundary conditions handled?
6. Can you provide physics-based validation metrics (spectra, etc.)?
soundness: 1
strengths: 1. **Novel architecture**: The ODE-SDE framework is interesting, explicitly incorporating advection dynamics while using neural SDEs for uncertainty quantification.
2. **Multi-model training**: Evaluation across 20 climate models demonstrates broader applicability than an emulator trained on a single model.
3. **Uncertainty quantification**: Training directly on NLL with multiple stochastic trajectories provides principled heteroscedastic uncertainty estimates.
4. **Parameter efficiency**: At 2.1M parameters, PACER is substantially smaller than state of the art emulators such as ACE. (However, I'm not convinced that these other models are prohibitively large.)
5. **Zero-shot generalization**: Table 3 demonstrates some cross-model transfer capability.
summary: This paper proposes PACER, a 2.1M parameter climate model emulator that combines a physics-informed ODE backbone with a neural SDE refinement module. The model is trained to predict surface temperature and precipitation fields from atmospheric GHG concentrations, using data from ClimateSet and FV3GFS. The authors demonstrate stable 10-year autoregressive rollouts and zero-shot generalization capabilities, outperforming selected UNet, ConvLSTM, and SFNO baselines on deterministic and probabilistic metrics.
weaknesses: 1. **Choice of physics prior**: Surface air temperature and precipitation are not advective processes. Indeed, in physics-based climate models, they're not prognostic variables at all, but rather are diagnosed. An advection equation therefore seems like a poorly motivated physics prior for variables that are not governed by a PDE.
2. **No computational timing results**: The paper's core motivation, and a primary motivation for climate model emulation generally, is computational efficiency. However, the paper provides no wall-clock time comparisons. Solving both an ODE and an SDE at each autoregressive step could be expensive. Without timing data, efficiency claims are unsupported.
3. **Physics consistency**: The authors claim that PACER is "physically consistent," but this claim is not supported. Beyond RMSE/CRPS, one might typically look conservation of mass/momentum/energy, but that isn't possible here (at least not for the ClimateSet dataset consisting of only surface temperature and precipitation). Other options would be: spatial power spectra, temporal autocorrelation functions, climate statistics.
4. **GHG forcing**: The $S(u, F; x, y, t)$ term in Eq. 3 is never actually specified. How are "the GHG forces" incorporated into the PDE? It's not obvious to me how this forcing would look.
5. **Unclear boundary condition handling**: No discussion of spherical geometry, for example, periodic boundaries and singularities at poles.
6. **Confidence intervals**: Tables present single numbers with no error bars, confidence intervals, or significance tests.
7. **Visual artifacts**: Figure 2 shows severe artifacts in PACER predictions that aren't addressed by the authors.
8. **Ensemble setup**: The ensemble mean at a given window serves as the input to the next window. The SDE then provides a measure of the spread within that window. However, this also means that the "ensemble" is not continuous across windows, that is to say, ensemble member $i$ in window $w$ does not follow from ensemble member $i$ in window $w-1$. I think this distinction, that PACER does not truly simulate an ensemble during autoregressive rollout, should be made very clear in the text.
9. **Velocity inference methodology (Section 4.3)**: This section is convoluted and poorly explained. The authors claim that "The standard approach is the climate-velocity index." I disagree that this approach is "standard" for approximating derivatives in general circulation models. Indeed, the cited paper (Gaponenko et al., 2022) aims to solve a different problem, that is, estimating the velocity of isotherms under climate change, which is distinct from the PDE being modeled in this paper. Also, the climate velocity should be $$v = \frac{1}{\|\nabla u\|}\frac{\partial u}{\partial t},$$ not $v =\|\nabla u\|\frac{\partial u}{\partial t}$ as it's written in the paper. This makes the units incorrect. Which version did you implement in practice?

---

confidence: 4
contribution: 3
initial_rating: 2
presentation: 1
questions: Was there any hyperparameter search that was done when running baselines on the FV3GFS dataset? Or did you reuse the parameters from ClimateSet?
soundness: 2
strengths: The idea of combining a physics-guided backbone with a stochastic residual corrector is a very compelling idea, that could be very useful in the field of climate modelling as uncertainty representation + physical consistency are two main goals. Previous research had proposed to use a physics-guided backbone with a deterministic residual corrector for weather modelling (Kochkov et al., 2023). 

Resolving the advection process with FDM makes sense, and the parameterization of the Neural SDE + Uncertainty quantification is sound.
summary: The paper proposes a climate model emulator that first resolves the advection equation, before refining the trajectory with a Neural SDE. The Neural SDE allows for accurate forecasting as well as uncertainty quantification, while incorporating the advection equation improves the skill of the model. 
The model is evaluated on two datasets: ClimateSet, a dataset containing TAS, PR, and forcings data from monthly-resolution climate models, and FV3GFS, which represents 7 levels of temperatures and radiative fluxes. It outperforms competing models in TAS RMSE. 

Overall, the paper shows an interesting model, with a physics-backbone, but I am not sure that the chosen physics backbone is suited for the task of climate modelling and the limited evaluation does not provide enough evidence that it is.
weaknesses: First, the advection diffusion model describes the evolution of gases in the atmosphere over weather timescales (i.e. high temporal resolution), but not over monthly timescale, where the dynamics of sea-surface temperature are mostly driven by the ocean. The diffusion of forcings at a monthly temporal resolution are not at all described by the advection diffusion model, but is rather dependent on the boundary conditions. Similarly, the climate-velocity index or "local migration speed" was originally defined for tracking climate change i.e. estimating the velocity between two 30-years periods (Gaponenko et al., 2022). Computing it from monthly data does not make much sense as it will be largely dependent on the initial conditions (it might depend on the season, for example). 
    
Although the idea of using a physics-backbone for climate model emulation is compelling, I don't think that the chosen backbone (advection equation) is suited for the task. Such a model (on monthly data) would probably be better suited for modelling ocean dynamics. 

The authors empirically show that the RMSE is lower when using the physics-backbone. However, the evaluation is very limited and this is not enough for proving that the advection equation truly represents the climate dynamics.   

First, the result in figure 3 does not show statistical significance. I recommend training and evaluating the models with multiple seeds + initial conditions to get error bars and statistical significance. This could also be used to evaluate CRPS for competing models (as is traditionnally done with climate ensembles). 

Second, when using the advection equation, the model is informed of the "initial velocity", which might lead to better MSE in the next months prediction only, and could lead to the results shown in figure 3. Using metrics that reflect long-term vs short-term prediction is necessary here. 

Third, RMSE is not by itself enough to verify whether a climate emulator performs well. It favors smooth models (double-penalty of MSE) but more importantly, the goal of climate modelling is to capture climate trends, as the exact temperature on a specific month is not predictable. Thus, I would recommend reporting statistics like the mean + std. dev., or the power spectral density of the climate indices as done in [1], or the time-dependent area-weighted global mean and the area-weighted global mean bias and RMSE of time-mean fields as done in [2]. 

Fourth, the models are not evaluated on precipitation, although there are only two output variables in climateset, temperature and precipitation. No performance on precipitation is reported, and the second dataset contains temperature at different levels. I understand that the goal is to emulate TAS, as a response to forcings, but emulating the interactions between variables is key for this task, and understanding the model's performance on precipitation, and whether or not the advection equation helps is very important. 

The model claims to perform uncertainty quantification. CRPS is reported but not compared. Also, no maps of the learned uncertainty is shown. It is also not clear if the uncertainty is well propagated during the autoregressive process. As mentioned above, baselines could be trained with multiple seeds and initial conditions to compute CRPS. 

The limited evaluation prevents from understanding the importance of the two model components, whether PACER actually outperforms baselines, and whether the choice of advection equation as a physics-backbone is supported by empirical evidence.  

Finally, the presentation of the paper could be improved. The zero-shot emulation section is hard to grasp ("We report the RMSE and CRPS for TAS (surface air temperature) pre-trained on different climate models"; no "pretrained climate model" column in table 3). Figure 2 illustrates "stronger fidelity", but does not show any comparison, and labels are small and blurry. In section 4 and Figure 1, it is not clear what u or x are (x,y are coordinates in the left part, but x is then the variable of interest in the right part of Figure 1?)

[1] Hickman et al., Causal Climate Emulation with Bayesian Filtering, 2025

[2] Watt-Meyer et al., ACE: A fast, skillful learned global atmospheric model for climate prediction, 2023

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confidence: 5
contribution: 2
initial_rating: 2
presentation: 1
questions: The RMSE scores in ClimateSet (fig. 6) seem to be significantly better than the ones reported here (Table 1). I understand that ClimateSet models it as a diagnostic-type task, while PACER is an autoregressive model, but are there any other contributors to this mismatch? Does this imply that diagnostic-type emulation is superior? If so, maybe the paper would benefit from a different perspective? In any case, simply omitting the existing results from climateSet without discussion is dissatisfying.
soundness: 1
strengths: - Physics-informed design for a climate emulator is a great direction that could benefit the emulators' trustworthiness and robustness
- Better RMSE scores than the deterministic baselines presented in the paper.
summary: The paper proposes PACER, a 2.1M parameter climate emulator designed for auto-regressive simulation. Its core contribution is a hybrid ODE-SDE framework, where a physics-based ODE models the advection-forcing dynamics , and a neural SDE refines the trajectory while capturing stochastic variability. The model predicts the mean and stddev of a normal distribution, and can thus be trained on a Gaussian NLL. The model demonstrates stable 10-year auto-regressive rollouts, at monthly resolution, and promising performance across 20 climate models.
weaknesses: 1. Sloppy writing hurts the flow of reading and makes the writing feel rushed. E.g., in the abstract: *"PACER is trained across is trained across"*, use proper upper-cases (ghg, unet, sde, pde...), lines 51-53 and 115-117, name SDE_integrate and sdint consistently, enocder -> encoder in Fig. 1, multiple surface levels is wrong (same issue in Table 2; only one level can be the surface level...) etc.
2. The proposed emulator is trained/tested on data at monthly resolution only. This is extremely coarse. For example, all the noted emulators in lines 96-102 run at 6 hourly resolution. The coarser resolution makes autoregressive modeling a much easier task (the model now needs to predict monthly averages instead of high-frequency weather). If the paper compared to one of these existing emulators and identified benefits for using monthly data, I'd be more inclined to let this pass.
3. Besides, PACER only emulates temperature. That's certainly a very important variable, but again, the existing emulators already go beyond that and emulate the whole atmosphere (e.g. humidity, winds, precip. etc.). Both the coarse temporal resolution and limited variable set reduce the practicability of the emulator, reduce problem complexity, and make it hard to compare it with existing approaches. 
4. Relatedly, the model learns a Gaussian fit to the data. That might work for variables like temperature but certainly not for many other important variables like precip. This needs to be mentioned as a limitation. Besides, none of the results are for non-temperature variables, even though precip. is part of the used ClimateSet dataset and the original FV3GFS data (as well as many more variables which were dropped in this paper without any explanation for why).
5. Strong artefacts in the predicted global maps (e.g., see Fig. 2) raise the question of how good this emulator actually is.
6. It's unclear how the reported RMSE/CRPS are computed. Are these timestep-wise (i.e. month by month) scores, or computed on time-mean's? 
7. No runtime benchmark, which is important to include for emulators.
8. The claim *"deterministic objective such as MSE/RMSE, which fits the conditional mean but leaves the predictive spread under-determined"* is inaccurate. Most diffusion models are trained on an MSE-based loss, but their posterior distribution is still quite strong.