# A variational neural network approach for glacier modelling with nonlinear rheology

arXiv: https://arxiv.org/abs/2209.02088 · HTML: https://mlsanity.com/paper/2209.02088/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 71 references that are in the map.
- First posted on arXiv 5 Sep 2022.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [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)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [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)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method](https://mlsanity.com/paper/2111.01356/) (2021)
- [Deep Multiscale Model Learning](https://mlsanity.com/paper/1806.04830/) (2018)
- [A Deep Learning Approach to Nonconvex Energy Minimization for Martensitic Phase Transitions](https://mlsanity.com/paper/2206.13937/) (2022)
- [Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics](https://mlsanity.com/paper/1406.6573/) (2014)

## Built on by (most cited first, 2 of 2)

- [PODNO: Proper Orthogonal Decomposition Neural Operators](https://mlsanity.com/paper/2504.18513/) (2025)
- [On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures](https://mlsanity.com/paper/2605.21388/) (2026)

## Abstract (arXiv)

In this paper, we propose a mesh-free method to solve full stokes equation which models the glacier movement with nonlinear rheology. Our approach is inspired by the Deep-Ritz method proposed in [12]. We first formulate the solution of non-Newtonian ice flow model into the minimizer of a variational integral with boundary constraints. The solution is then approximated by a deep neural network whose loss function is the variational integral plus soft constraint from the mixed boundary conditions. Instead of introducing mesh grids or basis functions to evaluate the loss function, our method only requires uniform samplers of the domain and boundaries. To address instability in real-world scaling, we re-normalize the input of the network at the first layer and balance the regularizing factors for each individual boundary. Finally, we illustrate the performance of our method by several numerical experiments, including a 2D model with analytical solution, Arolla glacier model with real scaling and a 3D model with periodic boundary conditions. Numerical results show that our proposed method is efficient in solving the non-Newtonian mechanics arising from glacier modeling with nonlinear rheology.