Redes neuronales de álgebra geométrica para la simulación de la ecuación de aguas poco profundas

Abstract

The Shallow Water Equations (SWE) are the standard mathematical framework for simulating free-surface geophysical flows; tsunamis, coastal flooding, and large-scale atmospheric and oceanic currents, yet their traditional numerical solution faces considerable computational bottlenecks, particularly over complex topographies. This work evaluates an alternative based on Geometric Algebra Neural Networks (RNAG), architectures that replace classical vector products with the Clifford geometric product, seeking an inductive bias that respects the vector and rotational structure of the flow. Three architectures were implemented and compared, a conventional convolutional network (BasicCNN), a physics-informed network (ShallowWaterPINN), and a geometric algebra network (ShallowWaterClifford), trained on a synthetic dataset generated with SpeedyWeather.jl, and evaluated both on one-step prediction and on the stability of autoregressive rollout simulation. Results show a non-trivial pattern: ShallowWaterPINN achieved the lowest one-step prediction error, in both vector and scalar fields, while ShallowWaterClifford, despite a less accurate one-step fit, produced the lowest rollout error among the three architectures. This reversal suggests that the geometric structure of Clifford Algebra does not provide an advantage in instantaneous accuracy, but may better contain error propagation under autoregressive composition, a property particularly relevant for simulators intended for sustained use. The comparatively poor performance of BasicCNN is explained by an early interruption of its training, rather than by an architectural limitation.