# A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling arXiv: https://arxiv.org/abs/2301.11402 ยท HTML: https://mlsanity.com/paper/2301.11402/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Builds on 11 of its 52 references that are in the map. - First posted on arXiv 26 Jan 2023. ## Builds on (most cited first, 11 of 11) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020) - [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021) - [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013) - [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022) - [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013) - [Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet](https://mlsanity.com/paper/1410.1221/) (2014) - [Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet](https://mlsanity.com/paper/2104.01996/) (2021) - [Constraining subglacial processes from surface velocity observations using surrogate-based Bayesian inference](https://mlsanity.com/paper/2006.12422/) (2020) - [Performance portable ice-sheet modeling with MALI](https://mlsanity.com/paper/2204.04321/) (2022) - [Variational Inference at Glacier Scale](https://mlsanity.com/paper/2108.07263/) (2021) ## Built on by (most cited first, 11 of 11) - [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023) - [Graph convolutional network as a fast statistical emulator for numerical ice sheet modeling](https://mlsanity.com/paper/2402.05291/) (2024) - [History-Aware Neural Operator: Robust Data-Driven Constitutive Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2506.10352/) (2025) - [JAX-MPM: A Learning-Augmented Differentiable Meshfree Framework for GPU-Accelerated Lagrangian Simulation and Geophysical Inverse Modeling](https://mlsanity.com/paper/2507.04192/) (2025) - [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023) - [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025) - [Physics-Informed Machine Learning On Polar Ice: A Survey](https://mlsanity.com/paper/2404.19536/) (2024) - [Emulation with uncertainty quantification of regional sea-level change caused by the Antarctic Ice Sheet](https://mlsanity.com/paper/2406.17729/) (2024) - [Graph Neural Network as Computationally Efficient Emulator of Ice-sheet and Sea-level System Model (ISSM)](https://mlsanity.com/paper/2407.01464/) (2024) - [A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results](https://mlsanity.com/paper/2407.17182/) (2024) - [Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets](https://mlsanity.com/paper/2512.01888/) (2025) ## Abstract (arXiv) One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics. At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models. In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces (parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model. We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.