The core of the GM09 model describes the change as a function of time (t, in years) in regolith thickness (h, in meters) as a balance between the regolith production rate from bedrock (Pr, in m yr⁻¹) and the physical erosion rate at the surface (E, in m yr⁻¹): 1dhdt=Pr-E Simultaneously, the model tracks the evolution of the fractional abundance of weatherable primary minerals (x, unitless) within the regolith. This concentration changes as a function of time (t, years) and depth (z, m) due to both upward advection with the regolith column and chemical dissolution: 2∂x∂t=-Pr∂x∂z-Kτσx where τ is the time a given rock particle has spent within the regolith, and Kτσ represents the dissolution rate constant, which depends on the local climate, exposure time (τ), and an empirical exponent (σ). The evolution of exposure time τ itself is tracked as: 3∂τ∂t=-Pr∂τ∂z+1 The net weathering rate for the entire regolith column (W, in m yr⁻¹) is the integral of the chemical dissolution term over the full regolith thickness: 4W=∫0hKτσxdz where z = 0 is at the base of regolith or the surface of bedrock.

The key rates in the model (Pr, E, and K) are parameterized as functions of environmental conditions. The production rate (Pr) is modeled as the product of an “optimal” production rate P0 and a soil-production function fh that describes how production decreases with increasing regolith thickness: 5Pr=P0fh where the optimal rate P0 (which occurs on bare bedrock) is dependent on runoff (q; unit: m yr⁻¹) and temperature (T; unit: K), following an Arrhenius relationship: 6P0=krpqeEaR1T0-1T Where Ea is the activation energy (J mol⁻¹), R is the ideal gas constant (8.314 J (mol K)⁻¹), T0 is the reference temperature (K), and krp is a proportionality constant.

The soil-production function f(h) takes an exponential form, where d0 is a reference regolith thickness (m): 7fh=e-hd0 In the GM09-SPIM model, the physical erosion rate is parameterized using the Stream Power Incision Model (SPIM): 8E=keqmSn where ke is a proportionality constant, S is the local slope (m m⁻¹), and m and n are exponents (typically 0.5 and 1). This SPIM formulation (Eq. 8) is the specific component that our new data-driven model replaces, as it is the primary source of the model's systematic biases.

The dependence of the chemical dissolution rate constant on climate is captured by an empirical function that includes the effects of both runoff and temperature: 9K=kd1-e-kwqeEaR1T0-1T where kd and kw are empirical proportionality constants.

While GM09 is a transient model, for long-term geological applications it is typically run to a steady-state solution where the regolith thickness, mineral concentration and exposure time are all constant (dh/dt=0, ∂x/∂t=0, ∂τ/∂t=0). Under this assumption, the regolith production rate equals the erosion rate (Pr=E) according to Eq. (1). The particle residence time at height z above the bedrock therefore simplifies to τ=z/E according to Eq. (3). Substituting these into the mineral mass balance (Eq. 2) yields KzEσx=-E∂x∂z. Integrating this rate throughout the entire regolith column (z=0 to z=h, Eq. 4) yields the total flux W=Exz=0-xz=h, where xz=0 and xz=h are the mineral concentrations at the bedrock and surface, respectively. Vertical integration of KzEσx=-E∂x∂z gives ∫xz=0xz=h1xdx=-KEσ+1∫0hzσdz, from which we have the solution xz=h=xz=0e-Kσ+1hEσ+1. Substituting the expression of K (Eq. 9), the weathering flux W simplifies to an expression: 10W=Exz=0-xz=0e-kd1-e-kwqeEaR1T0-1Tσ+1hEσ+1 This steady-state solution (Eq. 10) is central to our study. It explicitly demonstrates that the total weathering flux is the direct product of the physical erosion rate (E) and the total fraction of material that is chemically weathered. This highlights the paramount importance of accurately estimating E. An incorrect E will propagate directly into an incorrect W, which is precisely the problem we aim to solve.

According to Eq. (10), apart from the erosion module, weathering flux is also dependent on temperature and runoff. Mean annual temperature (T; K) is derived from CRU TS version 4.03 (Harris et al., 2014), while runoff (R; m yr⁻¹) is obtained from the Geophysical Fluid Dynamics Laboratory (GFDL) CM2.0 employed in Park et al. (2020). Zuo et al. (2024) demonstrated that these datasets provide robust performance when applied to silicate weathering models. Note that q is exactly the same as RUNOFF in the erosion module developed by Zhao et al. (2026), while T is the same as MAT in the erosion module but transformed to unit of kelvin.

Moreover, The Ca and Mg cation concentrations (x) depend on the type of rocks. Lithology is from the Global Lithologic Map (GLiM) (Hartmann and Moosdorf, 2012), which provides 16 categorical rock types. We classified the rocks into six groups, following the approach of Park et al. (2020) and Zuo et al. (2024) (see Table S2 in the Supplement). The Ca and Mg cation concentrations for each rock type can be obtained from the EarthChem database (http://portal.earthchem.org/, last access: 23 May 2024). Sedimentary and metamorphic rocks, however, exhibit properties that strongly depend on their protoliths, introducing significant uncertainty into the estimation of silicate weathering fluxes. To address this, Park20 treated the cation concentrations of these two rock categories as adjustable parameters in their model, and the same strategy is applied in this study.

Our observational “ground truth” for erosion is derived from the OCTOPUS database (Codilean et al., 2018), which compiles ¹⁰Be and other cosmogenic radionuclide (CRN) measurements from fluvial sediments (see Fig. 2 for data distribution and Supplement, Sect. S1.1 for technical details). We utilized the global dataset of ∼ 4100 basin-wide denudation rates (variable name is EBE_MMKYR, in mm kyr⁻¹) calculated from ¹⁰Be data using the CAIRN program (Mudd et al., 2016), assuming a standard rock density of 2650 kg m⁻³.

We used the base-10 logarithm of the denudation rate as the target variable for machine learning. This transformation serves two purposes. First, it converts the highly right-skewed distribution of denudation rates into a more normal distribution, which is better suited for most regression algorithms (see Fig. S1 in the Supplement). Second, it minimizes the influence of extreme outliers, enhancing model robustness.

The selection of predictor variables/features was guided by established geomorphic principles (Fig. 3). Because the OCTOPUS denudation rates are basin-averaged, all predictor variables were processed to represent basin-wide mean values. The sources, coverage extent and time window of these variables finally selected are provided in Table S1 and their spatial distribution are shown in Fig. S2.

Basin-mean slope (SLOPE; m km⁻¹) and elevation (ELEV; m) are calculated from the 0.5° × 0.5° topography map from Park20 directly, whose topography was from the Shuttle Radar Topography Mission (Farr et al., 2007). When the global erosion rates are calculated on grids, we also take the slope and elevation data from Park20 directly. This data is used so that the silicate weathering fluxes calculated can be compared to those in Park20.

Present-day mean annual temperature (MAT; °C) is from CRU TS v.4.03 (Harris et al., 2014) and runoff (RUNOFF; m yr⁻¹) is from the “Yves” dataset from Geophysical Fluid Dynamics Laboratory (GFDL) CM2.0 as used in Park et al. (2020), which were shown by Zuo et al. (2024) to yield strong performance in weathering models. Precipitation of the wettest month (PWET; mm) is from WorldClim 2 (Fick and Hijmans, 2017), and is proved to be one of the most important climatic variables that determines the erosion rate in the detachment-limited regime (Zhao et al., 2026). Leaf Area Index (LAI, unitless) is from NCAR, derived from integrated land observations (Lawrence and Chase, 2007). Lithology (ROCK, unitless) is from the Global Lithologic Map (GLiM) (Hartmann and Moosdorf, 2012), which provides 16 categorical rock types (see Table S2).

To obtain basin averages from the gridded climate, vegetation, and lithology data, we performed a spatial intersection using QGIS (a geographic information system software) so that each basin may be composed of partial grid cells. The value for each basin was calculated via area-weighted averaging for continuous variables (temperature, precipitation, runoff, LAI). For the categorical lithology variable, ROCK, we assigned a dominant rock type to a basin only if a single type covered > 70 % of its area; basins with more mixed lithologies were excluded to reduce ambiguity. After this filtering, our final dataset for training comprised 3721 valid samples, each with seven predictor variables: SLOPE, ELEV, MAT, PWET, RUNOFF, LAI, and ROCK. Sensitivity tests showed that reducing the number of predictors below seven (e.g., removing the least important ROCK) led to a consistent decrease in both erosion (R2 dropped from 0.85 to 0.78) and weathering (R2+Rlog2 dropped from 0.86 to 0.72) model performance.

We also tested the inclusion of peak ground acceleration (PGA; m s⁻²), which is an indicator of local tectonic activities. However, as it provided only a marginal improvement in model performance (ΔR2 ≈ 0.017) and is difficult to reconstruct for geological history, it is excluded from the final model. Mean Diurnal Range (MDR; °C) is also excluded in this model because of the limited reconstruction ability in paleoclimate. Hence, another difference between the model here and that in Zhao et al. (2026) is that less predictors are used herein. Vegetation is represented using LAI rather than Normalized Difference Vegetation Index (NDVI) and the Enhanced Vegetation Index (EVI), because it is a more common output from earth system models. We also tested the effect of other climatic variables that were used in the model of Zhao et al. (2026), such as mean annual precipitation (MAP; mm yr⁻¹), seasonality of temperature (TSEASON; °C; the standard deviation of monthly temperature) and precipitation (PSEASON; defined as the ratio of the standard deviation of monthly mean to the annual mean precipitation rate, in unit %), temperature of the coldest month (TMIN; °C), all collected from WorldCLim 2 (Fick and Hijmans, 2017). However, adding these variables did not improve but deteriorate the performance on the calculation of both erosion rates and weathering flux when using 0.5° × 0.5° data, especially the latter (Table 1), and thus not considered in this study.

We compared several supervised regression algorithms, including linear models, decision trees, gradient boosting, and Random Forest. While decision trees offer excellent interpretability, Random Forest consistently yielded the highest predictive accuracy (see Supplement, Sect. S1.2 and Fig. S3 for details). The Random Forest (RF) algorithm (Breiman, 2001) is an ensemble model that builds a multitude of decision trees and aggregates their predictions. It is particularly well-suited for environmental data due to its ability to handle complex non-linearities and its robustness to multicollinearity (e.g., the correlation between latitude, elevation, and temperature).

The dataset was randomly split into a training set (80 %) and a testing set (20 %). We performed a grid search with 10-fold cross-validation on the training set to identify the optimal combination of hyperparameters. The search grid included: ntree (number of trees): [300, 500]; mtry (number of variables to sample at each split): [2, 5]; nodesize (minimum samples in a terminal node): [1, 5]; maxdepth (maximum tree depth): [10, 30]. The optimal combination that minimized RMSE was found to be: ntree = 500, mtry = 2, nodesize = 1, maxdepth = 30. The final RF model was trained on the full training set using these parameters and evaluated on the unseen testing set.

To understand the drivers learned by the model, the importance of each predictor or feature variable is ranked based on the TreeSHAP method. TreeSHAP is a variant for tree-based machine learning models of SHapley Additive exPlanations (SHAP; Lundberg and Lee, 2017), which is based on coalitional game theory (Shapley, 1953). TreeSHAP scores (positive/negative or null) of each feature are the changes in the expected model prediction due to that feature. Specifically, it is the average change in the model's output when this feature is varied over all possible values, while keeping all other features constant.

The trained RF model was used to predict a global map of erosion rates (ERF) at 0.5° × 0.5° resolution. This data-driven erosion map was then used as the direct input for the erosion rate E in the GM09 weathering model framework. We refer to the original model using the SPIM erosion law as GM09-SPIM, and our improved version with the new erosion map as MErSiM.

To provide a robust and fair comparison, both the GM09-SPIM and MErSiM models were run using identical boundary conditions, including the same GLiM lithology map and climate forcing fields. The model outputs, generated at a 0.5° × 0.5° resolution were evaluated against established observational datasets.

Our primary validation dataset for silicate weathering is the widely-used compilation by Gaillardet et al. (1999). This dataset provides observation-based estimates of silicate-derived Ca²⁺ and Mg²⁺ fluxes for 52 of the world's largest river basins. These fluxes are inferred from river water chemistry using an inverse method to deconvolve sources (atmospheric, carbonate, silicate, anthropogenic), which is particularly effective for large, lithologically complex catchments. We therefore benchmark MErSiM against the Ca + Mg silicate-derived flux, not against total silicate-derived alkalinity or total CO₂ consumption by all cations.

This choice imposes two important boundaries on the current model-data comparison. First, using Ca + Mg flux as a long-term carbon-cycle metric assumes that the relevant Ca + Mg-derived alkalinity is ultimately balanced mainly by carbonate burial. In deep-time intervals with strong reverse weathering, authigenic clay formation, or other non-carbonate alkalinity sinks, users should prescribe the oceanic partitioning of alkalinity rather than treating the MErSiM output alone as a complete carbon sink. Second, basaltic weathering is included in MErSiM but is not reported by the Gaillardet et al. (1999) large-river compilation. It should be noted that MErSiM v1.0 represents basaltic rocks in the GM09 framework with a high mafic Ca + Mg dissolution parameter, rather than through a dedicated empirical basalt-weathering law such as that proposed by Dessert et al. (2003). The detailed discussion of basaltic weathering is in the Supplement, Sect. S1.3.

Zuo et al. (2024) noted a significant discrepancy in the Amazon basin between the weathering flux reported by Gaillardet et al. (1999) (∼ 0.02 mol m⁻² yr⁻¹) and that from the HYBAM (Hydrogeochemistry of the Amazon Basin) observatory (∼ 0.07 mol m⁻² yr⁻¹), which consists of 32 smaller, well-monitored sub-catchments (Moquet et al., 2011, 2016). Instructed by the result of Zuo et al. (2024), we validate MErSiM using a combined dataset where the Amazon basin data from Gaillardet is replaced by the HYBAM data. Following community practice (Park et al., 2020; Zuo et al., 2024), we focus our main analysis on the 81 non-overlapping major basins from the compilation of Gaillardet et al. (1999) and HYBAM.

We assessed the agreement between modeled and observed fluxes for each basin using the coefficient of determination (R2). Following Zuo et al. (2024), we consider both the standard R2 (calculated on direct values) and the R2 calculated on log₁₀-transformed values (Rlog2). The former is sensitive to biases in total flux, while the latter better reflects the model's ability to capture the correct order of magnitude across diverse basins. We also use R2+Rlog2 as the final evaluating metric as Zuo et al. (2024). Note that R2+Rlog2 was also maximized for GM09-SPIM by searching the model parameter space (see Table 2 for the parameters that are searched), in addition to the model parameters previously fitted by Park et al. (2020). By using this comprehensive evaluation framework, which includes multiple observational benchmarks, we can rigorously quantify not only if our new model (MErSiM) is better, but specifically how and where it improves upon the reference model GM09-SPIM.

To ensure that our calibrated parameter set represents a robust optimal solution rather than an arbitrary point in a poorly constrained parameter space, we performed a comprehensive sensitivity analysis. Figure S4 visualizes slices of the high-dimensional parameter space, illustrating how model performance (quantified by the R2+Rlog2 metric) varies with changes in key parameters. The results demonstrate that our optimization has converged on a well-defined and constrained solution. Across the plots for the primary kinetic parameters (σ,kw,krp, Fig. S4a–c), a distinct high-performance region (the bright yellow “sweet spot”) emerges, centered around our selected optimal values (indicated by the red boxes). The plots for lithological parameters (Fig. S4d–e) reveal that while the model performance is highly sensitive to the choice of kd, it is relatively insensitive to a wide range of cation concentrations for both metamorphic and sedimentary rocks, as long as kd is within its optimal range.

A common criticism of machine learning models is that they can be “black boxes”, providing accurate predictions without revealing the underlying physical mechanisms (Breiman, 2001). To address this and build confidence in MErSiM's physical realism, we employed SHAP (SHapley Additive exPlanations) analysis. SHAP is a state-of-the-art method that assigns each feature an importance value for every individual prediction, allowing us to move beyond general rankings and understand the nuanced, non-linear effects of each environmental driver. We also calculate the spearman correlation of erosion rates and each feature. Figures 6 and 7 provides insights into which environmental factors the model found to be most important and the nature of their relationships with erosion.

Figure 6a displays the global feature importance, calculated as the mean absolute SHAP value across all training samples. SLOPE emerges as the unequivocally dominant predictor, with a mean SHAP value of 0.6, nearly double that of the next most important feature. This finding is geomorphologically reassuring, as it aligns with geomorphic research and first-principles understanding (Milliman and Syvitski, 1992; Portenga and Bierman, 2011; Riebe et al., 2001), and also consistent with previous data-driven analysis (Portenga and Bierman, 2011). It confirms that our data-driven model has independently learned that topography is the master variable controlling the pace of erosion at a global scale.

Following slope, a group of climatic variables – MAT (Mean Annual Temperature) and RUNOFF – exhibit significant and comparable importance (SHAP values of 0.39 and 0.34, respectively). This highlights that after the primary topographic driver, the model relies heavily on climatic conditions to refine its predictions. Note that the influence of RUNOFF is secondary after SLOPE and even MAT, a contrast to the SPIM, where runoff is a dominant, first-order driver. The RF model has learned that high runoff only leads to high erosion when other conditions, primarily steep slopes, are met (see below).

LAI (Leaf Area Index) follows as a moderately important factor, underscoring the role of vegetation in modulating erosional processes. PWET (Precipitation of the Wettest Month) and ELEV (Elevation) show lower, yet still relevant, importance, while ROCK (Lithology) has the least impact, suggesting that at a global scale, broad rock types are less predictive than the dynamic interplay of topography and climate.

This order of importance, where slope ranks the first with significantly higher SHAP value, then following climatic variables with similar importance, and rock type ranks the last, is well consistent with that in Zhao et al. (2026), even when they have 14 predictor features included.

The Spearman correlation plot (Fig. 6b) shows strong positive correlations between erosion rate and SLOPE, and moderate positive correlations with ELEV and RUNOFF, indicating that steeper, higher, and wetter environments generally erode faster. MAT displays a negative correlation with erosion rate.

To move beyond general feature rankings and understand the nonlinear effects of each predictor, we utilize SHAP dependence plots (Fig. 7). Each point on a SHAP dependence plot represents a single sample (i.e., a river basin) from our dataset. The horizontal axis indicates the actual value of the feature for that sample, while the vertical axis shows its corresponding SHAP value. The SHAP value itself quantifies the magnitude and direction of a feature's contribution to a single prediction, relative to the baseline (mean) prediction across all samples. Its units are the same as the model's output – in our case, logarithmic erosion rate. Here, our baseline is 4.4 (equivalent to an erosion rate of ∼ 78 mm kyr⁻¹). This is the model's best prediction without any specific information. If we provide the model with the specific features of predictors: for instance, one of river basins in Dadu, SLOPE = 13.6°, MAT = 6.7 °C, RUNOFF = 0.75 m yr⁻¹, and so on. The model processes these inputs and yields a final prediction, 6.1 (equivalent to ∼ 438 mm kyr⁻¹). The SHAP values explain precisely how the model arrived at this prediction by decomposing the difference between the final prediction (6.1) and the baseline (4.4). Each feature is assigned a SHAP value representing its individual contribution: 11Baseline+Σ(SHAP values for all features)=Final Prediction In this case, SHAPSLOPE=0.91, SHAPMAT=0.14, SHAPRUNOFF=0.34, SHAPLAI=0.28, SHAPPWET=0.03, SHAPELEV=-0.01, SHAPROCK=0.03, then: 124.4+0.91+0.14+0.34+0.28+0.03-0.01=4.4+1.7=6.1

The resulting distribution of points thus reveals the overall relationship – be it linear, non-linear, monotonic, or more complex – between a feature's magnitude and its impact on the model's output.

The SHAP value of SLOPE (Fig. 7a) is negative for gentle slopes (< ∼ 3°), indicating that flat landscapes actively suppress erosion predictions. Above this threshold, the SHAP value increases steeply and almost linearly up to about 10–15°, after which the effect begins to saturate. This saturation is physically realistic and consistent with a shift from transport-limited to detachment-limited regimes in the steepest landscapes, a nuance that simple power-law models like SPIM cannot capture (Binnie et al., 2007; Ouimet et al., 2009).

For MAT (Fig. 7b), the model has learned a complex relationship. When MAT is below ∼ 10 °C, SHAP values are positive and peak at ∼ 0 °C (i.e., cold temperatures push erosion predictions higher and this effect is the strongest around 0 °C). This is consistent with the “frost-cracking window” hypothesis, which posits that erosion is maximized not in the absolute coldest environments, but in those that experience frequent oscillations across the 0 °C threshold (Anderson and Anderson, 2010; Delunel et al., 2010). These freeze-thaw cycles generate substantial mechanical stress within bedrock fractures, leading to efficient rock breakdown and the production of a large volume of transportable sediment, even in the absence of steep topographic gradients (Hales and Roering, 2007). Above ∼ 10 °C, warmer temperatures are associated with lower erosion predictions, and this lowering effect remains the strongest around 20–25 °C. This negative trend in warmer climates likely captures multiple effects, including the stabilization of landscapes by tropical leached soil (deeply weathered, erosion-resistant regoliths) as noticed before by Zuo et al. (2024).

The effect of RUNOFF (Fig. 7c) is highly non-linear and exhibits a clear optimal peak. Erosion predictions increase with runoff up to about 1.7 m yr⁻¹, after which higher runoff values lead to a decrease in predicted erosion. It suggests the model has learned that moderate runoff in transport-capable landscapes is the most effective to activate hydrological erosion (Carretier et al., 2013; DiBiase and Whipple, 2011). However, the decrease of variability in discharge with the increase of runoff under wetter conditions (e.g., > 2 m yr⁻¹) would result in a suppressed net impact on erosion (DiBiase and Whipple, 2011; Lague, 2014; Perron, 2017). The asymmetric hump-shaped relationship has also been reported and discussed in detail in the statistical analysis of Zhao et al. (2026). This data-driven insight directly refutes the monotonic, power-law assumption of the SPIM.

The SHAP plot for LAI (Fig. 7d) reveals its dual role. At very low LAI values (< ∼ 0.5), it has a slight positive effect, likely reflecting arid or semi-arid environments where sparse vegetation is insufficient to stabilize mobile sediment. As LAI increases from ∼ 1.5 to ∼ 2.5, it exerts a strong negative influence on erosion, capturing the well-established stabilizing effect of root systems and canopy cover (Mishra et al., 2019; Prosser et al., 1995). Interestingly, at very high LAI values (> 3), the negative impact lessens. This could reflect the fact that the most densely vegetated regions on Earth are often flat, stable cratons where erosion is already intrinsically low, so additional vegetation has a diminishing marginal effect (Heimsath et al., 1997).

The SHAP dependence plot of the least three variables (PWET, ELEV and ROCK) show their less significant effects on erosion rates and more ambiguous trends (Fig. S5), so the analyses of their effects are only briefly described in the Supplement.

A representative decision tree from the Random Forest ensemble (Fig. 8) offers a powerful visualization of the complex hierarchical relationships learned by the model. The tree structure reveals that the importance of different variables is conditional and depends on the specific environmental context.

The very first split at the top of the tree is based on SLOPE, with a threshold of 2.5° (calculated from DEM with 0.5° × 0.5°). This immediately separates the world into low-relief and high-relief domains, indicating that the fundamental drivers of erosion differ profoundly between these two settings.

In the flattest parts of the world (SLOPE < 2.5°), the model turns to MAT. This is also seen in the analysis of Zhao et al. (2026) in their model with more predictors. In these areas, the effect of river incision is weaker and the rate of erosion may be more strongly influenced by the rate of chemical decomposition or by freeze-thaw cycle (Anderson and Anderson, 2010; Delunel et al., 2010; White and Blum, 1995).

LAI became an important predictor as the slope steepened (2.5° ≤ SLOPE < 6.3°), and the data-driven model calculated an LAI threshold of 2. This value is exactly the same as the threshold used for the modification of the erosion model in Zuo et al. (2024), where the value was obtained by human judgement. Vegetation can prevent erosion by increasing surface roughness, disrupting overland water flow, and stabilizing the soil through the root system (Mishra et al., 2019; Prosser et al., 1995). The overall weak negative correlation between LAI and erosion rates and the branches in the decision tree suggest that vegetation primarily serves to reduce erosion.

In steep landscapes (SLOPE >= 6.3°), the model identifies RUNOFF as the next most important variable, with thresholds of 0.27 m yr⁻¹ (Fig. 8). This suggests that in very steep terrain, there is an abundant supply of material, and the rate of regolith detachment becomes limited by the availability of water to move it (DiBiase and Whipple, 2011; Snyder et al., 2003; Tucker, 2004).

Consistent with our modern-day simulations, the MErSiM model predicts significantly lower absolute global weathering fluxes across all climate cases compared to the Park20 model (Table 3). The PI Ca and Mg silicate weathering flux in MErSiM is 3.11 × 10¹² mol yr⁻¹, approximately 2/3 of the 4.54 × 10¹² mol yr⁻¹ predicted by the Park20 model, bringing the weathering fluxes approximately within the range of observation-based estimates (Gaillardet et al., 1999; Moon et al., 2014).

More importantly, our revised model exhibits a markedly different sensitivity to climate change. From LGM to PI, the weathering flux in MErSiM increases by 0.99 × 10¹² mol yr⁻¹, a relative increase of 47 %. This is comparable to the 46 % increase predicted by the Park20 model, suggesting that both models capture a similar weathering response to glacial-interglacial warming (Fig. 11a). However, the response to intense warming in the 4×CO₂ scenario diverges dramatically. While the Park20 model predicts a massive increase in weathering of 6.77 × 10¹² mol yr⁻¹ (a 149 % increase from PI), our MErSiM model predicts a much weaker response of only 1.33 × 10¹² mol yr⁻¹ (a 42 % increase, Fig. 11b). This profound attenuation of weathering sensitivity under warm, high-CO₂ conditions is a key finding of our study and points to a shift in the weathering regime captured by our improved erosion module.

The dampened response of MErSiM stems from a combination of a structural change in the model's physics and a corresponding shift in its calibrated chemical parameters.

The primary reason for the reduced sensitivity is the replacement of the simplistic SPIM erosion law with our data-driven, machine learning-based module. The SPIM creates a strong, near-universal positive coupling between runoff and erosion. Consequently, in a 4×CO₂ world with enhanced runoff, the Park20 model simulates a widespread, dramatic increase in erosion, continuously supplying fresh material and allowing chemical weathering to accelerate freely with rising temperatures.

Our ML-based erosion model, constrained by ¹⁰Be data, imposes a ceiling on erosion rates. It has learned that in many landscapes – particularly those with low-to-moderate slope that constitute the bulk of Earth's land area – erosion is strongly constrained by topography and vegetation, and does not accelerate dramatically with increased runoff alone (Fig. 7a, c). Moreover, erosion rates are overall negatively correlated with mean annual temperatures (Fig. 6b), especially in the flattest areas, where higher temperatures tend to make it more difficult for freeze-thaw cycles to occur and erosion rates to decrease (see decision tree in Fig. 8). As a result, when faced with the strong warming and hydrological intensification of the 4×CO₂ scenario, the acceleration of erosion due to enhanced runoff is largely offset, or even reversed, by the overall suppression of erosion by elevated temperatures, notably in the higher latitudes beyond 50° (Fig. 11d). Weathering fluxes are limited by the supply of erosion, effectively decoupling weathering from the large increase in chemical weathering potential due to temperature increase.

The second reason is a new kinetic pattern reflected by calibrated parameters. To match modern observational constraints within this erosion framework, MErSiM calibration converged on a set of chemical kinetic parameters that describe a far more “sluggish” weathering system than that of Park20 (Table 2). The most significant changes are (1) a 5-fold decrease in the base dissolution rate constant (kd) (1×10-4 vs. 5×10-4), lowering the intrinsic speed of chemical reactions; (2) a 5-fold decrease in the runoff sensitivity parameter (kw) (0.2 vs. 1.0), making chemical weathering less responsive to increases in water availability. (3) a qualitative shift in the weathering time-exponent (σ) from -0.4 to -0.1, implying a slower decay of reaction rates as minerals reside in the regolith.

These parameters are not arbitrary. The model has discovered that to explain today's weathering fluxes in a world where erosion is not as dynamic as previously assumed, the chemical processes themselves must be inherently slower and less sensitive. This finding is supported by several lines of evidence and theoretical work. A long-standing challenge in geochemistry is the dramatic discrepancy, often spanning several orders of magnitude, between mineral dissolution rates measured in the laboratory and those observed in the field (White and Brantley, 2003). The 5-fold lower dissolution constant (kd) value derived from our calibration reflects a large-scale effective rate compared to the rapid rates seen on fresh mineral surfaces under ideal laboratory conditions. This rate is naturally attenuated by processes such as the formation of passivating secondary mineral coatings, the reduction of reactive surface area over time, and the complexities of fluid flow paths within the regolith.

This finding can also be explained by the hydrologic regulation framework of Maher and Chamberlain (2014), which posits that weathering fluxes are governed by the competition between fluid travel time (τf) and the time required for reactions to approach equilibrium (τeq). Our results can be interpreted through the lens of their Damköhler number formulation. The high dissolution constant (kd) in the Park20 model implies a very short intrinsic equilibrium time (τeq), suggesting a system that can rapidly approach its “thermodynamic limit”. This allows weathering fluxes to scale aggressively with increased runoff, particularly in high-relief areas. In contrast, the 5-fold lower kd value and altered kinetic exponent (σ) derived from our calibration describe a system with a much longer intrinsic τeq. According to Maher and Chamberlain (2014), weathering fluxes plateau and become limited when fluid travel times become short relative to the equilibrium time (τf<τeq). In a warmer world with a greatly intensified hydrologic cycle, runoff increases, leading to a global decrease in τf. For the Park20 model, with its inherently short τeq, the system relatively remains in a state where τf>τeq, and weathering fluxes continue to climb even with shorter time for chemical reaction. However, for our revised model, this widespread decrease in τf causes vast terrestrial areas to be pushed into the regime where equilibrium could not be fully established. In this state, solute concentrations decrease as runoff increases, and the overall weathering flux saturates. The lower runoff sensitivity (kw=0.2) in our revised model is a direct macro-scale expression of this saturation effect. Our model has learned from the data that, on a global scale, many landscapes already operate closer to this kinetically limited boundary than previously assumed. Consequently, under the intense hydrological forcing of a 4×CO₂ world, they cannot deliver a proportionally large increase in weathering flux.

This attenuated sensitivity is reinforced by the model's negative temperature-erosion coupling. As temperatures rise significantly, certain landscapes – particularly those previously dominated by frost-cracking – experience a reduction in physical denudation rates (Anderson and Anderson, 2010; Zhao et al., 2026). This reduction in the supply of fresh minerals imposes a secondary ceiling on the chemical weathering potential, even as reaction kinetics are accelerated by warming.

For 4×CO₂ world, it is important to recognize that highly reactive components of the Earth system may provide more rapid transient adjustments to climate forcing. For instance, volcanic basaltic terrains exhibit a much higher temperature sensitivity than average silicate rocks (Li et al., 2016). Furthermore, while our focus remains on the Ca + Mg alkalinity flux for long-term carbon burial, the rapid response of Na and K silicate weathering can serve as a critical transient CO₂ buffer. The potential saturation of the continental terrestrial sink, as identified by MErSiM, suggests that maintaining global carbon cycle mass balance (Berner and Caldeira, 1997) under extreme greenhouse conditions may require more dynamic contributions from marine shelves (Trapp-Müller et al., 2025), and seafloor basalts (Coogan and Gillis, 2018).