# Physics-enriched neural solvers for transient ice-flow simulation

arXiv: https://arxiv.org/abs/2609.12900 · HTML: https://mlsanity.com/paper/2609.12900/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 45 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
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- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomization](https://mlsanity.com/paper/2505.12149/) (2025)
- [Performance portable ice-sheet modeling with MALI](https://mlsanity.com/paper/2204.04321/) (2022)

## Abstract (arXiv)

Transient glacier simulations with higher-order ice flow require the repeated solution of a nonlinear problem as the geometry evolves. In the online mode of the Instructed Glacier Model, the velocity field is represented by a neural network whose weights are warm-started from the previous time step and updated with a few optimizer iterations. We show that supplying the network with inexpensive input fields derived from low-order ice-flow balances improves this online solver. Unlike residual-based physics-informed neural networks, which incorporate physics through governing-equation penalties in the loss, our approach leaves the governing energy objective unchanged, adding physical structure through the network inputs. Across three real-world glacier configurations, the enriched solver is markedly more robust to solver settings. On the two alpine cases, it also improves the tuned accuracy--runtime trade-off, reducing surface-velocity errors by factors of two to four at fixed runtime and reaching few-percent relative errors with only 104--105 trainable parameters, far fewer than comparable raw-input baselines. A 300-year Aletsch simulation then completes in under one minute, and the larger Valais domain in about two minutes, on a single GPU---a budget once reserved for much simpler shallow-ice models. Gains are smaller for the fast marine-terminating glacier, where nonlocal stress coupling favors larger or spectral networks. More broadly, the results suggest that enriching a neural solver's inputs with reduced-order physics can make repeated higher-order solves much cheaper, with no training data and no offline training.