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Abstract. Subglacial topography beneath the Greenland Ice Sheet is a fundamental control on its dynamics and response to changes in the climate system. Yet, it remains challenging to measure directly, and existing representations of the subglacial topography rely on a limited number of observations. Although the use of mass conservation and the development of BedMachine Greenland substantially improved the representation of the bed topography, this approach is limited to fast-flowing sectors and is less effective in regions with complex, alpine topography. As an alternative to traditional numerical methods, recent work has explored using Physics Informed Neural Networks (PINNs), constrained by only one physical law, to solve forward and inverse problems in ice sheet modeling. Building on this work, we assess three PINN frameworks constrained by distinct conservation laws, showing that PINNs informed with a single conservation law are not sufficient for regions with sparse measurements and complex topographies. To that end, we introduce a novel approach that involves coupling two conservation laws within a PINN framework to infer the subglacial topography and test this approach for three regions with distinct environments in Greenland. This PINN is trained with both the conservation of mass and an approximation of the conservation of momentum (the Shelfy-Stream Approximation), which allows us to simultaneously infer the ice thickness and basal shear stress using observations of ice velocities, surface elevation, and the apparent mass balance in a mixed inversion problem. We compare the predicted ice thickness to ground-truth ice-penetrating radar measurements of ice thickness, showing that the PINN informed with two conservation laws is capable of inferring ice thickness in sparsely surveyed regions. Furthermore, comparisons of predicted bed topographies with BedMachine Greenland show that this approach is capable of discovering new bed features in slower-moving regions and in regions of complex topography, highlighting its potential for better constraining the bed topography of the Greenland Ice Sheet. 

Reconstructing past climate from moraine records is complicated by the influence of non-climatic factors, particularly topography, on glacier extent. Such topographic controls have been widely identified in the literature, but a systematic quantitative assessment of their effects on glacier extent is lacking. Here, we investigate the relative influence of topographic and climatic factors on tropical glacier length variability in the Sierra Nevada del Cocuy, Colombia using a coupled ice-flow–energy-balance glacier model. Employing a parameter sweep over 450 topographic scenarios and 40 climatic scenarios for a total of 18,000 unique topo-climatic scenarios, we identify a critical transition in glacier length around 5 °C to 6 °C below modern temperature where variability in inter-valley glacier length shifts from headwall elevation-controlled to valley slope-controlled. We show through a relative weights analysis that, for this particular topo-climatic parameter space, climate accounts for 84% of the modeled variability in glacier length, while topography contributes 16%. Among climatic variables, temperature plays a more dominant role than precipitation, and headwall elevation influences glacier length most of any topographic variable. After accounting for all possible combinations of parameter subsets, we find that a sizable portion of topo-climatic scenarios (22%) yields glacier lengths dominated by topographic factors rather than climatic factors. These findings highlight the complex interplay between climate and topography, demonstrating that topography, though typically secondary to climate, has a notable impact on glacier length in this particular glacier regime. As such, this study provides a framework for quantifying the relative contributions of climate and topography to glacier evolution, critical for interpreting past glacier extents and predicting future changes. 

Predicting the future contributions of the ice sheets to sea level rise remains a significant challenge due to our limited understanding of key physical processes (e.g., basal friction, ice rheology) and the lack of observations of critical model inputs (e.g., bed topography). Traditional numerical models typically rely on data assimilation methods to estimate these variables by solving inverse problems based on conservation laws of mass, momentum, and energy. However, these methods are not versatile and require extensive code development to incorporate new physics. Moreover, their dependence on data alignment within computational grids hampers their adaptability, especially in the context of sparse data availability in space and time. To address these limitations, we developed PINNICLE (Physics-Informed Neural Networks for Ice and CLimatE), an open-source Python library dedicated to ice sheet modeling. PINNICLE seamlessly integrates observational data and physical laws, facilitating the solution of both forward and inverse problems within a single framework. PINNICLE currently supports a variety of conservation laws, including the Shelfy-Stream Approximation (SSA), Mono-Layer Higher-Order (MOLHO) models, and mass conservation equations, for both time-independent and time-dependent simulations. The library is user-friendly, requiring only the setting of a few hyperparameters for standard modeling tasks, while advanced users can define custom models within the framework. Additionally, PINNICLE is based on the DeepXDE library, which supports widely-used machine learning packages such as TensorFlow, PyTorch, and JAX, enabling users to select the backend that best fits their hardware. We describe here the implementation of PINNICLE and showcase this library with examples across the Greenland and Antarctic ice sheets for a range of forward and inverse problems. 

Abstract. Geologic archives of the Laurentide Ice Sheet (LIS) provide abundant constraints regarding the size and extent of the ice sheet during the Last Glacial Maximum (LGM) and throughout the deglaciation. Direct observations of LGM LIS thickness are non-existent, however, due to ice surface elevations likely exceeding those of even the tallest summits in the northeastern United States (NE USA). Geomorphic and isotopic data from mountains across the NE USA can inform basal conditions, including the presence of warm- or cold-based regimes, while covered by ice. While warm-based ice and erosive conditions likely existed on the flanks of these summits and throughout neighboring valleys, cosmogenic nuclide inheritance and frost-riven blockfields on summits suggest ineffective glacial erosion and cold-based ice conditions. Geologic reconstructions indicate that a complex erosional and thermal regime likely existed across the NE USA sometime during and after the LGM, although this has not been confirmed by ice sheet models. Instead, current ice sheet models simulate warm-based ice conditions across this region, with disagreement likely arising from the use of low-resolution meshes (e.g., > 20 km) which are unable to resolve the high bedrock relief across the NE USA that strongly influenced overall ice flow and the complex LIS thermal state. Here, we use a newer-generation ice sheet model, the Ice-sheet and Sea-level System Model (ISSM), to simulate the LGM conditions of the LIS across the NE USA and in three localities with high bedrock relief (Adirondack Mountains, White Mountains, and Mount Katahdin), with results confirming the existence of a complex thermal regime as interpreted from the geologic data. The model uses a small ensemble of LGM climate boundary conditions and a high-resolution horizontal mesh that resolves bedrock features down to 30 m to reconstruct LGM ice flow, ice thickness, and thermal conditions. These results indicate that, across the NE USA, polythermal conditions existed during the LGM. While the majority of this domain is simulated to be warm-based, cold-based ice persists where ice velocities are slow (< 15 m yr−1), particularly across regional ice divides (e.g., Adirondack Mountains). Additionally, sharp thermal boundaries are simulated where cold-based ice across high-elevation summits (White Mountains and Mount Katahdin) flanks warm-based ice in adjacent valleys. We find that the elevation of this simulated thermal boundary ranges between 800–1500 m, largely supporting geologic interpretations that polythermal ice conditions existed across the NE USA during the LGM; however, this boundary varies geographically. In general, we show that a model using a finer spatial resolution compared to older models can simulate the polythermal conditions captured in the geologic data, with the model output being of potential utility for site selection in future geologic studies and for geomorphic interpretations of landscape evolution. 

Predicting the future contributions of the ice sheets to sea-level rise remains a significant challenge due to our limited understanding of key physical processes (e.g., basal friction, ice rheology) and the lack of observations of critical model inputs (e.g., bed topography). Traditional numerical models typically rely on data assimilation methods to estimate these variables by solving inverse problems based on conservation laws of mass, momentum, and energy. However, these methods are not versatile and require extensive code development to incorporate new physics. Moreover, their dependence on data alignment within computational grids hampers their adaptability, especially in the context of sparse data availability in space and time. To address these limitations, we developed PINNICLE (Physics-Informed Neural Networks for Ice and CLimatE), an open-source Python library dedicated to ice sheet modeling. PINNICLE seamlessly integrates observational data and physical laws, facilitating the solution of both forward and inverse problems within a single framework. PINNICLE currently supports a variety of conservation laws, including the Shelfy-Stream Approximation (SSA), MOno-Layer Higher-Order (MOLHO) models, and mass conservation equations, for both time-independent and time-dependent simulations. The library is user-friendly, requiring only the setting of a few hyperparameters for standard modeling tasks, while advanced users can define custom models within the framework. Additionally, PINNICLE is based on the DeepXDE library, which supports widely used machine learning packages such as TensorFlow, PyTorch, and JAX, enabling users to select the backend that best fits their hardware. We describe here the implementation of PINNICLE and showcase this library with examples across the Greenland and Antarctic ice sheets for a range of forward and inverse problems. 

Abstract. The Greenland Ice Sheet's negative mass balance is driven by a sensitivity to both a warming atmosphere and ocean. The fidelity of ice-sheet models in accounting for ice-ocean interaction is inherently uncertain and often constrained against recent fluctuations in the ice-sheet margin from the previous decades. The geological record can be utilised to contextualise ice-sheet mass loss and understand the drivers of changes at the marine margin across climatic shifts and previous extended warm periods, aiding our understanding of future ice-sheet behaviour. Here, we use the Ice-sheet and Sea-level System Model (ISSM) to explore the Holocene evolution of Ryder Glacier draining into Sherard Osborn Fjord, Northern Greenland. Our modelling results are constrained with terrestrial reconstructions of the paleo-ice sheet margin and an extensive marine sediment record from Sherard Osborn Fjord that details ice dynamics over the past 12.5 ka years. By employing a consistent mesh resolution of <1 km at the ice-ocean boundary, we assess the importance of atmospheric and oceanic changes to Ryder Glacier's Holocene behaviour. Our simulations show that the initial retreat of the ice margin after the Younger Dryas cold period was driven by a warming climate and the resulting fluctuations in Surface Mass Balance. Changing atmospheric conditions remain the first order control in the timing of ice retreat during the Holocene. We find ice-ocean interactions become increasingly fundamental to Ryder's retreat in the mid-Holocene; with higher than contemporary melt rates required to force grounding line retreat and capture the collapse of the ice tongue during the Holocene Thermal Maximum. Regrowth of the tongue during the neo-glacial cooling of the late Holocene is necessary to advance both the terrestrial and marine margins of the glacier. Our results stress the importance of accurately resolving the ice-ocean interface in modelling efforts over centennial and millennial time scales, in particular the role of floating ice tongues and submarine melt, and provide vital analogous for the future evolution of Ryder in a warming climate. 

Abstract Predicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we use the Physics‐Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two‐dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g., ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes. 

Abstract. Geologic evidence of the Laurentide Ice Sheet (LIS) provides abundant constraints on the areal extent of the ice sheet during the Last Glacial Maximum (LGM). Direct observations of LGM LIS thickness are non-existent, however, with most geologic data across high elevation summits in the Northeastern United States (NE USA) often showing signs of inheritance, indicative of weakly erosive ice flow and the presence of cold-based ice. While warm-based ice and erosive conditions likely existed on the flanks of these summits and throughout neighboring valleys, summit inheritance issues have hampered efforts to constrain the timing of the emergence of ice-free conditions at high elevation summits. These geomorphic reconstructions indicate that a complex erosional and thermal regime likely existed across the southeasternmost extent of the LIS sometime during the LGM, although this has not been confirmed by ice sheet models. Instead, current ice sheet models simulate warm-based ice conditions across this region, with disagreement likely arising from the use of low resolution meshes (e.g., >20 km) which are unable to resolve the high bedrock relief across this region that strongly influenced overall ice flow and the complex LIS thermal state. Here we use a newer generation ice sheet model, the Ice-sheet and Sea-level System Model (ISSM), to simulate the LGM conditions of the LIS across the NE USA and at 3 localities with high bedrock relief (Adirondack Mountains, White Mountains, and Mount Katahdin), with results confirming the existence of a complex thermal regime as interpreted by the geologic data. The model uses higher-order physics, a small ensemble of LGM climate boundary conditions, and a high-resolution horizontal mesh that resolves bedrock features down to 30 meters to reconstruct LGM ice flow, ice thickness, and thermal conditions. These results indicate that across the NE USA, polythermal conditions existed during the LGM. While the majority of this domain is simulated to be warm-based, cold-based ice persists where ice velocities are slow (<15 m/yr) particularly across regional ice divides (e.g., Adirondacks). Additionally, sharp thermal boundaries are simulated where cold-based ice across high elevation summits (White Mountains and Mount Katahdin) flank warm-based ice in adjacent valleys. Because geologic data is geographically limited, these high-resolution simulations can help fill gaps in our understanding of the geographical distribution of the polythermal ice during the LGM. We find that the elevation of this simulated thermal boundary ranges between 800–1500 meters, largely supporting geologic interpretations that polythermal ice conditions existed across NE USA during the LGM, however this boundary varies geographically. In general, we show that a model with finer spatial resolution and higher order physics is able to simulate the polythermal conditions captured in the geologic data, with model output being of potential utility for site selection in future geologic studies and geomorphic interpretation of landscape evolution.