[PAOCQ] Doug Brinkerhoff (University of Montana)

“Why my glacier flow emulator turned back into a PDE solver: GPU-accelerated multigrid, ML-inspired transport operators, and the difficulties that necessitated them.”

Machine-learning surrogates for ice dynamics are often motivated by the high computational cost of traditional flow models. Here, I describe a somewhat unexpected outcome of pursuing that approach: attempting to build neural-network emulators ultimately led back to classical numerical methods—albeit with several ideas borrowed from modern machine learning.

Initial work focused on graph-based neural operators and diffusion-style architectures designed to emulate shallow-shelf ice dynamics. These models emphasized properties that are also desirable in physical solvers:  rotational and scale invariance, discretization independence, and information transport mechanisms that do not rely on fixed receptive fields. While such architectures produced promising emulators, the design process revealed strong parallels with well-established numerical techniques.

These insights ultimately motivated the development of Glide, a GPU-accelerated ice-dynamics solver built around nonlinear geometric multigrid and discretization-invariant transport operators. Many structural features of modern neural architectures have direct numerical analogues: multigrid V-cycles resemble U-Net hierarchies, restriction and prolongation correspond to pooling and upsampling, and stencil-based flux operators act as physics-consistent convolutional layers.  Unlike neural emulators, however, this approach preserves exact physical constraints, delivers predictable convergence properties, and is amenable to implicit time-stepping schemes. As with PINNs and surrogates, the model remains differentiable for inverse problems through adjoint methods, while being less expensive to differentiate through time because of the lack of hidden state.

The resulting solver achieves orders-of-magnitude performance gains through GPU-native implementation while retaining the robustness of classical PDE methods. More broadly, this work suggests that the interaction between machine learning and scientific computing need not always produce neural surrogates—sometimes the most productive outcome is a re-examination of classical algorithms through the lens of modern ML architectures.

PAOC Colloquium —

Interdisciplinary seminar series that brings together the whole PAOC (Program in Atmospheres, Oceans, and Climate) community. Seminar topics include all research concerning the physics, chemistry, and biology of the atmospheres, oceans and climate, as well as talks about societal impacts of climatic processes.

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