Glacier retreat, driven by anthropogenic climate change, poses significant environmental and societal challenges. Around 10% of the world's population relies on glacier-fed water resources for various purposes, including agriculture, hydropower, and domestic use. Predicting future glacier evolution is crucial for mitigating these impacts. Existing global glacier evolution models often employ temperature-index models, which utilize a linear relationship between positive degree-days (PDDs) and melt. While these models are suitable for large-scale studies with limited data, they offer only a linearized approximation of the inherently nonlinear climate-glacier interactions. Deep artificial neural networks (ANNs), capable of capturing nonlinear relationships, present an alternative. However, their use in glaciology for regression problems remains largely unexplored. The French Alps, experiencing significant glacier retreat, offer a unique case study due to long-term monitoring data and societal dependence on glacier resources. This study investigates the nonlinear response of French Alpine glaciers to future climate change using a novel deep learning framework, which integrates a deep learning mass balance component and glacier-specific parametrizations. The climatic forcing is derived from high-resolution climate ensemble projections, encompassing 29 combinations of global climate models (GCMs) and regional climate models (RCMs) across three Representative Concentration Pathway (RCP) scenarios (2.6, 4.5, and 8.5). This research provides novel projections of glacier evolution in a densely populated mountain region and examines the impact of nonlinearities in glacier response to multiple climate forcings.

Numerous studies have highlighted the global scale of glacier mass loss and its consequences for sea level rise and water resources. Global glacier evolution models have been developed to estimate future sea-level contributions, but these models often rely on simplifying assumptions, particularly in representing glacier mass balance. Temperature-index models, despite their simplicity and suitability for large-scale studies, are known to have limitations due to their linear approximation of the complex interactions between climate variables and glacier melt. Some previous research has explored the use of artificial neural networks in glaciology, primarily for tasks like predicting ice thickness or mass balance of individual glaciers. However, the application of deep learning to project regional-scale glacier evolution remains largely uncharted territory. The existing literature indicates that the French Alps have seen considerable glacier retreat and represent a significant area of study due to extensive historical glaciological data and associated societal impact. The studies on glacier evolution projections in the European Alps have been conducted mostly using temperature-index models, making it crucial to assess their accuracy for varied climate scenarios and different glacier types.

This study employs a novel modeling framework, the Alpine Parameterized Glacier Model (ALPGM), integrating a deep learning component for mass balance estimation and glacier-specific parametrizations for geometry changes. Glacier-wide mass balance (MB) is simulated annually for individual glaciers using two approaches: a deep artificial neural network (deep learning) and the Lasso (regularized multilinear regression). This methodology was detailed in a previous publication. The ALPGM uses a feed-forward fully connected multilayer perceptron with a specific architecture (40-20-10-5-1), activation functions, and an optimization strategy to avoid overfitting. The Lasso model serves as a linear counterpart for comparison. A dataset of 32 glaciers with direct annual glacier-wide MB observations and remote sensing estimates from 1967-2015 was used for model training and validation. Topographical and climate predictors were compiled for these glacier-year values. Topographical predictors include glacier altitude, area, slope, latitude, longitude, and aspect. Climate predictors comprise annual cumulative positive degree-days (CPDD), winter and summer snowfall, and monthly temperature and snowfall. The models were rigorously cross-validated using Leave-One-Glacier-Out (LOGO), Leave-One-Year-Out (LOYO), and Leave-Some-Years-and-Glaciers-Out (LSYGO) techniques to assess performance in spatial, temporal, and spatiotemporal dimensions. Simulations for projections were generated by an ensemble of 60 cross-validated models to improve robustness. Future projections utilized climate projections from ADAMONT, a statistically adjusted EURO-CORDEX dataset specific to French mountain regions, for 29 different future climate scenarios (2005-2100) across three RCPs. Glacier geometry evolution was parameterized using empirical functions to redistribute annual mass changes across glaciers, accounting for ice dynamics and MB. This parametrization was individually computed for glaciers larger than 0.5 km² using two DEMs (1979 and 2011). For smaller glaciers, mass changes were assumed to be uniform. Model performance was validated by comparing simulated glacier surface areas to observations from a 2015 glacier inventory. Simulations were conducted with varying initial ice thickness and geometry parameters to estimate uncertainties. A sensitivity analysis using a deterministic sampling process compared the nonlinear deep learning model with the linear Lasso model, assessing their responses to CPDD, winter snowfall, and summer snowfall anomalies. A comparison was also performed against GloGEMflow, a global glacier evolution model, requiring adjustments to account for differences in climate data, ice thickness estimates, and model parameters. A MB bias correction was applied to GloGEMflow's output for improved comparability.

The deep learning projections indicate substantial glacier mass loss across all climate scenarios, with average ice volume losses by 2100 ranging from 75% under RCP 2.6 to 88% under RCP 8.5. Differences between scenarios become more pronounced in the second half of the century. Annual glacier-wide mass balance (MB) is projected to remain relatively stable under RCP 4.5 but become increasingly negative under RCP 8.5. Under RCP 2.6, MB rates approach equilibrium towards the end of the century. Summer air temperature is identified as the primary driver of glacier mass change, while changes in annual snowfall are less pronounced due to higher precipitation rates at higher altitudes and glacier retreat to higher elevations. The analysis of the mass balance models reveals significant nonlinearities in the relationship between climate variables (CPDD, winter and summer snowfall) and MB. Deep learning shows a much stronger r² of 0.76 compared to Lasso's 0.41 in cross-validation. The linear Lasso model is found to be under-sensitive to extreme positive CPDD and over-sensitive to extreme negative CPDDs, and over-sensitive to extreme positive and negative snowfall anomalies, particularly summer snowfall. This highlights the limitations of linear approximations in representing extreme MB values. The topographical feedback from glacier retreat to higher elevations mitigates the effect of warming on mountain glaciers, particularly under higher emission scenarios. However, a synthetic experiment simulating ice cap-like behavior (no topographical adjustment) demonstrates that the absence of this feedback leads to a greater frequency of extreme negative MB rates and magnified differences between the nonlinear and linear models. Comparison with a large-scale glacier evolution model (GloGEMflow) that uses a temperature-index MB model indicates similar patterns. Both the Lasso and the temperature-index model show a tendency to overestimate positive MB rates under RCP 2.6 and underestimate mass loss under the lower emission scenario. The differences between deep learning and linear models were greater for flatter glaciers.

The findings highlight the limitations of temperature-index models, widely used in large-scale glacier projections, due to their linear assumptions. These models, calibrated on past data, fail to accurately capture the nonlinear response of glaciers to extreme future climate conditions. This oversensitivity stems from the changing influence of energy fluxes in the glacier surface energy budget with increasing temperatures. The nonlinear deep learning model more accurately reflects this reduced sensitivity, especially during the ablation season when ice is exposed. The study's results suggest that the biases introduced by linear models are less pronounced for mountain glaciers under high-emission scenarios due to the topographical adjustment. However, for flatter glaciers (ice caps), the lack of topographical feedback leads to more significant biases across all climate scenarios. The comparison with GloGEMflow reinforces the observed patterns, indicating that temperature-index models may overestimate mass balance under low-emission scenarios and underestimate it for high-emission scenarios. The implications of these biases for long-term projections are significant, potentially leading to underestimations of sea-level rise under low-emission scenarios and overestimations under high-emission scenarios.

This study demonstrates the advantages of using deep learning for regional-scale glacier MB modeling, accurately capturing nonlinearities and improving the representation of extreme MB rates. The limitations of linear MB models, including temperature-index models, are highlighted, especially for projections under extreme climatic conditions. The importance of topographical feedback for mountain glacier response is emphasized, in contrast to the behavior of flatter glaciers and ice caps. These findings raise concerns about the accuracy of existing large-scale glacier projections and underscore the need for improved models that capture nonlinear climate-glacier interactions. Future research should focus on developing nonlinear parameterizations and bridging the gap between domain-specific equations and machine learning.

The study focuses on the French Alps, which may not be fully representative of all glacierized regions. The synthetic experiment simulating ice cap behavior is a simplification and does not fully account for all processes affecting ice caps, such as thinning and interactions with oceans. The model's accuracy relies heavily on the quality of the input data, particularly climate projections and initial ice thickness estimates. Deep learning models, while powerful, can be treated as black boxes, and a complete understanding of the underlying physical processes might require further investigation.