cs.LGSep 25, 2026
SaveLearning Hierarchical Causal Representations of the Effects of Forcings on Temperature in Climate Models
Organizations: Technical University of Munich Munich, Germany · Karlsruhe Institute of Technology Karlsruhe, Germany · McGill University & Mila Montreal, Canada · Intel Labs Israel
Abstract
Machine learning (ML) emulators provide a fast and cost-effective method to simulate climate change scenarios after being trained on Earth System Models projections. However, the black-box nature of those data-driven approaches limit the usability and trustworthiness of their outputs and in particular their use as causal attribution tools. Here, we develop a hierarchical causal representation learning framework applied to sea surface temperature fields from a state-of-the-art global climate model. As a key advance over previous work, our framework explicitly models both atmospheric dynamical interactions arising from internal climate variability and forced responses due to changes in atmospheric greenhouse gas and aerosol concentrations. When trained on future climate change scenarios, our method accurately predicts the long-term global mean and regional temperature evolution and shows physically realistic responses to perturbations in greenhouse gas and aerosol concentrations when evaluated on unseen scenarios. Our results underline the potential of causal representation learning frameworks for advancing climate model emulation.
Figures & tables
Figure 1 : Hierarchical climate emulator. A global latent captures climate responses to global forcings, while local latents capture regional dynamics conditioned on spatially localized aerosols under the single-parent mapping. The model supports multi-scenario emulation and counterfactual perturbations of forcings.
Figure 2 : Long-term climate rollouts and forcing perturbations. (a) Our GMST rollouts under SSP3-7.0 (blue) compared with NorESM2 (black) and Linear Pattern Scaling (LPS) prediction (orange). Fixing CO 2 and CH 4 at their 2015 levels (red) suppresses the long-term increase in GMST. (b) Power spectral density of the Niño3.4 index under SSP3-7.0. The model ensemble mean (red) reproduces the characteristic Niño3.4 spectral peak. Shading in (a,b) denotes the std. across 50 trajectories. Details of the ablations and comparison to LPS are in C.2 . (c,d) Next-step temperature responses to regional BC and SO 2 perturbations over East Asia, showing warming and cooling responses, respectively. Green boxes indicate the perturbed region.
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3 : Overview of the hierarchical generative model. At each timestep, greenhouse-gas forcings (CO 2 and CH 4 ) condition the global latent , whose temporal evolution depends on its previous states. The resulting global state conditions the local latents , which additionally depend on their sparse temporal parents and spatially localized aerosol forcings (BC and SO 2 ). Aerosol forcing is assigned to each local latent according to the learned latent-to-observation mapping, while each observation is generated from its associated local latent under the single-parent structure. This hierarchy enables global greenhouse-gas responses and regional aerosol effects to enter the climate dynamics through different spatial levels.
| Parameter | Value |
|---|---|
| Latent representation | |
| Number of spatial grid cells | 3072 |
| Number of local latent variables | 48 |
| Number of global latent variables | 1 |
| Temporal context length | 5 months |
| Encoder and decoder | |
Table 1 : Model and training hyperparameters used in the experiments. We denote an MLP with hidden dimension and layers as , and use analogously for the corresponding linear layer. The model is trained with RMSProp ( , batch size ) for up to iterations. The auxiliary loss term incorporates the continuous ranked probability score (CRPS) and a spatio-temporal spectral loss on the predicted temperature.
Figure 4 : GMST rollouts and forcings. (a) GMST rollouts under SSP1-2.6 (blue), SSP2-4.5 (red), and SSP3-7.0 (green). Predicted GMST (dashed) is compared with NorESM2 (solid), with shading indicating standard deviations across 50 trajectories. Our model is able to distinguishes the scenario-dependent warming trends and captures the stronger warming under the held-out SSP3-7.0 scenario. (b) CO 2 , spatially averaged CH 4 , BC, and spatially averaged SO 2 forcings are shown for SSP1-2.6, SSP2-4.5, and SSP3-7.0 over the 85-year evaluation period. SSP1-2.6 and SSP2-4.5 represent lower- and intermediate-forcing pathways, respectively, while SSP3-7.0 represents a scenario with higher greenhouse gas and aerosol concentrations.
Figure 5 : Spherical power spectral density (PSD) of the time-averaged global surface temperature field. For each trajectory, monthly surface temperature maps are first averaged over time, followed by a spherical harmonic decomposition of the resulting global HEALPix map. The spatial power spectrum is represented by the angular power spectrum . The horizontal axis shows the approximate angular scale, , where smaller values correspond to finer spatial structures. The black curve denotes the NorESM2 reference, while the red curve shows the median PSD across model trajectories, with the shaded region indicating the 16th–84th percentile range. Both spectra are dominated by large-scale spatial structures, with power decreasing toward finer scales. The model closely reproduces the overall spectral slope and scale hierarchy of NorESM2, while it overestimates the power at fine spatial scales.
Figure 6 : GMST across 85 year rollouts under SSP3-7.0. The left panel shows the global mean surface temperature (GMST) over the full 85-year autoregressive rollout. The right panel provides a detailed view of the first 50 months, showing the ensemble mean and standard deviation across 50 trajectories, together with all 50 individual sample trajectories.
Figure 7 : Niño3.4 index across 85-year rollouts under SSP3-7.0. The left panel shows the Niño3.4 index over the full 85-year autoregressive rollout. The right panel provides a detailed view of the first 50 months, showing the mean and standard deviation of 50 trajectories, and 50 individual sample trajectories.
Figure 8 : GMST time-series autocorrelation under SSP3-7.0. The left panel shows the autocorrelation of GMST over the full 85-year autoregressive rollout, comparing the model prediction with the NorESM2 ground truth. The right panel provides a detailed view of the first 50 months, highlighting the short-term temporal dependence and seasonal oscillations. Both time series are linearly detrended prior to computing the autocorrelation to remove the long-term warming trend and prevent it from dominating the autocorrelation at large time lags. The model captures the approximately annual oscillatory structure of the reference autocorrelation, while exhibiting weaker short-term persistence.
Figure 9 : Niño3.4 time-series autocorrelation under SSP3-7.0. The left panel shows the autocorrelation of the Niño3.4 index over the full 85-year autoregressive rollout, comparing the model prediction with the NorESM2 ground truth. The right panel provides a detailed view of the first 50 months, highlighting the short-term temporal dependence and oscillatory variability. Both time series are linearly detrended prior to computing the autocorrelation to remove the long-term trend and prevent it from dominating the autocorrelation at large time lags. Compared with GMST, the model more closely reproduces the temporal dependence of the Niño3.4 index at one-year lag.
Figure 10 : Power spectral density (PSD) of the GMST and Niño3.4 index time series under SSP3-7.0. The left panel shows the PSD of the GMST time series, while the right panel shows the PSD of the Niño3.4 index. The model PSD is computed across 50 autoregressive rollout trajectories, with the solid red line indicating the ensemble mean and the shaded region indicating standard deviation. The black line denotes the NorESM2 ground-truth spectrum. In the GMST panel, the blue line additionally shows the PSD of the learned global climate latent, illustrating its temporal variability across different time scales. All time series are linearly detrended prior to PSD estimation. Global latent, compared with the spatial mean of the predicted temperature field, is better at capturing the long-term variability of GMST. The model reproduces the major spectral peaks of the Niño3.4 index, including the annual peak and variability at approximately 2 to 5 year periods.
| GMST | Niño3.4 | |||||
| Mean | Std. Dev. | Range | Mean | Std. Dev. | Range | |
| Ground truth | 1.2403 | 0.5771 | 2.5144 | 1.3662 | 0.7751 | 4.2305 |
| Linear Pattern Scaling Lütjens et al. [2025] | 1.4711 | 0.7009 | 2.4247 | 1.5677 | 0.7383 | 2.5535 |
| Ours-No Localized Aerosols | 1.3281 | 0.5327 | 2.7091 | 1.4013 | 0.6702 | 4.3710 |
| Ours-No | 1.0015 | 0.6247 | 4.8010 | 1.1235 | 0.6663 | 4.7029 |
| Ours | 1.4107 | 0.6573 | 3.2710 | 1.4692 | 0.8205 | 4.7190 |
Table 2 : Statistics of long-term climate rollouts under SSP3-7.0 forcing . Starting from the climate state on 2015-01-01, each model is rolled out autoregressively for 85 years, conditioned at each time step on the corresponding SSP3-7.0 external forcings. The particle selection follows 2.2 . The left section shows the mean, standard deviation, range of the GMST, and the same for the Niño3.4 index on the right. The first row shows the reference statistics computed from the ground-truth NorESM2 SSP3-7.0 simulation, followed by comparison method, and then ablations. For all metrics, being closer to the ground truth is better.
Figure 11 : Ablation study of the temporal power spectral density (PSD) under SSP3-7.0, shown on a logarithmic scale. (a) GMST and (b) Niño3.4 PSDs from 85-year autoregressive rollouts. We compare the full model with variants without the local aerosol mapping and without the global-latent alignment loss . Removing amplifies temporal variability, especially over the low-frequency part. Removing the local mapping produces a smoother PSD and fails to capture the characteristic Niño3.4 peak.
Figure 12 : Sample visualization of the temperature response to aerosol perturbations. The upper row shows five different SO 2 perturbation levels applied within East Asia, and the lower row shows the corresponding next-step temperature predictions. Increasing SO 2 produces a cooling response, whereas increasing BC produces a warming response, with the strongest changes concentrated around the perturbed region.
Figure 13 : Single-step response to CO 2 perturbations. We evaluate the model under controlled CO 2 perturbations while keeping the input climate state and all other forcing variables fixed. Starting from a random selected initial condition, CO 2 is varied across its range of values observed under SSP3-7.0, and a single-step prediction is performed for each forcing value. (a) Relationship between the perturbed CO 2 concentration and the predicted global climate latent ( ). (b) Relationship between the predicted global latent and the corresponding predicted GMST. (c) Direct relationship between the perturbed CO 2 concentration and predicted GMST. Dashed lines indicate linear fits, with the Pearson correlation coefficient ( ), -value, and fitted slope reported in each panel. The results show a nearly linear response of the global latent to CO 2 perturbations, which is subsequently reflected in the predicted GMST.
Figure 14 : Single-step response to CH 4 perturbations. Compared with CO 2 , CH 4 exhibits a weaker modulation of the learned global latent, and this variation is propagated even more weakly to the next-step GMST prediction, resulting in a substantially weaker direct CH 4 –GMST response. The cooling response to increased CH 4 reflects a limitation of the current model in disentangling the effects of strongly correlated global forcings and remains an area for further improvement.
Figure 15 : Single-step response to BC perturbations. In contrast to the global greenhouse-gas forcings, BC shows almost no modulation of the learned global latent ( ), while exhibiting a clearer positive response in the predicted GMST ( ). This suggests that the modeled BC temperature response is primarily mediated through the local climate latents rather than the global latent.
Figure 16 : Single-step response to SO 2 perturbations. Similar to BC, SO 2 shows almost no modulation of the learned global latent ( ), while exhibiting a weak negative response in the predicted GMST ( ). The global latent responds to global forcing perturbations but remains insensitive to aerosol perturbations, providing evidence that the model separates global and localized forcings through distinct causal pathways.
Figure 17 : Global latent response to forcing perturbations in long-term rollout. Global latent trajectories under SSP3-7.0 with original forcings (black) and with CO 2 and CH 4 fixed at their 2015 levels (red), showing a substantially suppressed long-term increase under the perturbation.
Figure 18 : Temperature response to interventions on the predicted global climate latent. We directly intervene on the predicted global latent mean ( ) by setting it to , , , , and , while keeping the input climate history and external forcing conditions unchanged. For each intervention, the prescribed global latent is passed to the local climate transition model, and the resulting local latent state is decoded into the predicted surface temperature field. Each panel shows the prediction for the same input sample under a different value of the global latent. As the global latent increases, the predicted temperature field exhibits a progressively stronger warming response over much of the globe. This sensitivity experiment indicates that the learned global latent modulates the large-scale temperature response.