Neural Solvers for Ice-Flow Simulation

Transient glacier simulations now reach higher-order accuracy in under a minute, a significant performance gain over traditional grid-based methods that resolve complex stress coupling iteratively. This speedup comes from a physics-informed neural network that operates with no pre-existing training data or offline training phase. Instead of treating the network as a black box to solve non-linear differential equations from scratch, the architecture incorporates domain knowledge by augmenting the input layer with fields derived from the Shallow Ice Approximation (SIA).

Injecting Physics into the Input

Traditional ice-flow models rely on solving partial differential equations, which become computationally expensive when accounting for higher-order stress coupling in moving ice. The new approach uses a small neural network, containing between 10,000 and 100,000 trainable parameters, to represent the velocity field.

Rather than forcing the network to predict the entire velocity field from random initialization, the model injects physical knowledge directly into the input. The input layer concatenates spatial coordinates (x, y, z) with velocity estimates calculated from the SIA—a computationally inexpensive model that provides a first-order flow estimate based on local surface slope and thickness.

During the forward pass, the network processes these SIA values alongside coordinates. Because the network architecture is structured to map the input space to a correction, the model does not learn the full physics from zero. Instead, the SIA output serves as a 'prior' that biases the initial weights toward a physically plausible state. The network learns a mapping function that computes the 'delta' or high-order correction needed to satisfy the full momentum balance. By shifting the objective from predicting the absolute velocity to predicting the error between the SIA baseline and the accurate solution, the optimizer avoids searching the entire parameter space, converging rapidly on a much smaller, localized set of corrections.

Predictive Performance and Constraints

In tests on alpine glacier configurations, the physics-enriched solver reduced surface-velocity errors significantly compared to the baseline SIA. A 300-year simulation of the Aletsch glacier completes in under one minute on a single GPU. The mechanism succeeds because the input enrichment effectively pre-conditions the gradient-based search. By pinning the network to a baseline that satisfies the momentum balance at a lower order, the optimizer is constrained to search for the specific refinement required for higher-order precision.

This architecture also enables efficient warm-starting. Because the geometry of a glacier evolves slowly, the solution found at time 't' provides an excellent starting weight distribution for time 't+1'. The model essentially performs a series of localized updates rather than solving a new global problem at every time step.

For someone building in this space, the primary constraint is the reliance on temporal and physical continuity. The method relies on the assumption that the SIA baseline remains a close approximation of the true physics. In marine-terminating glacier scenarios where non-local stress coupling is dominant, the SIA provides insufficient information. The inputs fail to capture the long-range forces, and the network struggles to derive the correct velocity field. If a user tries to apply this to a system where the physics changes abruptly or in ways not captured by the local approximation, the warm-started weights will fail to converge. This forces a manual reset of the optimization process, which negates the performance advantages of the method. The model is currently optimized for these specific, structured geometry constraints; it is not a general-purpose differential equation solver.