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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. 

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. 

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 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. Many floating ice shelves in Antarctica buttress the ice streams feeding them, thereby reducing the discharge of icebergs into the ocean. The rate at which ice shelves calve icebergs and how fast they flow determines whether they advance, retreat, or remain stable, exerting a first-order control on ice discharge. To parameterize calving within ice sheet models, several empirical and physical calving “laws” have been proposed in the past few decades. Such laws emphasize dissimilar features, including along- and across-flow strain rates (the eigencalving law), a fracture yield criterion (the von Mises law), longitudinal stretching (the crevasse depth law), and a simple ice thickness threshold (the minimum thickness law), among others. Despite the multitude of established calving laws, these laws remain largely unvalidated for the Antarctic Ice Sheet, rendering it difficult to assess the broad applicability of any given law in Antarctica. We address this shortcoming through a set of numerical experiments that evaluate existing calving laws for ten ice shelves around the Antarctic Ice Sheet. We utilize the Ice-sheet and Sea-level System Model (ISSM) and implement four calving laws under constant external forcing, calibrating the free parameter of each of these calving laws by assuming that the current position of the ice front is in steady state and finding the set of parameters that best achieves this position over a simulation of 200 years. We find that, in general, the eigencalving and von Mises laws best reproduce observed calving front positions under the steady state position assumption. These results will streamline future modeling efforts of Antarctic ice shelves by better informing the relevant physics of Antarctic-style calving on a shelf-by-shelf basis. 

Abstract Differentiable Earth system models (ESMs) enable powerful applications such as sensitivity analysis, gradient‐based calibration, state estimation, boundary flux inversions, uncertainty quantification, and online machine learning. Reverse‐mode automatic differentiation (AD) efficiently provides gradients for such tasks, yet models have rarely included this capability because of complex, bespoke numerical algorithms. As part of theDifferentiable programming in Julia for Earth system modeling (DJ4Earth)initiative, we present improved capabilities of the AD tool Enzyme.jl and the new compiler transpilation tool Reactant.jl, augmented by sophisticated checkpointing algorithms, which, together make general‐purpose AD tractable and efficient for full‐fledged ESM components written in Julia. Operating at the low‐level virtual machine intermediate representation or multi‐level intermediate representation compiler levels, these frameworks support mutable memory, custom kernels, and compiler optimizations before and after differentiation. Julia‐specific challenges related to just‐in‐time compilation and garbage collection are handled efficiently. Reactant further enables automatic performance portability across central processing units, graphics processing units, and tensor processing units, facilitating use of emerging AI‐customized high‐performance computing architectures. We demonstrate these frameworks on four Julia‐based ESM components featuring diverse spatial discretizations and numerical algorithms: the rotating‐sphere shallow water model ShallowWaters.jl, the finite‐volume ocean model Oceananigans.jl, the finite‐element ice sheet model DJUICE.jl, and the spectral atmospheric model SpeedyWeather.jl. Across these ESM components, our tools compute efficient and correct gradients. These results establish a foundation for differentiable, high‐performance and performance‐portable ESMs that can integrate neural networks for unresolved processes, trained online, enabling next‐generation hybrid physics–machine learning ESMs constrained by physical dynamics and observations. 

Many floating ice shelves in Antarctica buttress the ice streams feeding them, thereby reducing the discharge of icebergs into the ocean. The rate at which ice shelves calve icebergs and how fast they flow determine whether they advance, retreat, or remain stable, exerting a first-order control on ice discharge. To parameterize calving within ice sheet models, several empirical and physical calving “laws” have been proposed in the past few decades. Such laws emphasize dissimilar features, including along- and across-flow strain rates (the eigencalving law), a fracture yield criterion (the von Mises law), longitudinal stretching (the crevasse depth law), and a simple ice thickness threshold (the minimum thickness law), among others. Despite the multitude of established calving laws, these laws remain largely unvalidated for the Antarctic Ice Sheet, rendering it difficult to assess the broad applicability of any given law in Antarctica. We address this shortcoming through a set of numerical experiments that evaluate existing calving laws for 10 ice shelves around the Antarctic Ice Sheet. We utilize the Ice-sheet and Sea-level System Model (ISSM) and implement four calving laws under constant external forcing, calibrating the free parameter of each of these calving laws for each ice shelf by assuming that the current position of the ice front is in steady state and finding the set of parameters that best achieves this position over a simulation of 200 years. We find that, in general, the eigencalving and von Mises laws best reproduce observed calving front positions under the steady-state position assumption. These results will streamline future modeling efforts of Antarctic ice shelves by better informing the relevant physics of Antarctic-style calving on a shelf-by-shelf basis.