Learning Debris Flow Dynamics with a Deep Learning Fourier Neural Operator: Application to the Rendinara–Morino Area

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Abstract
Accurate numerical simulation of debris flows is essential for hazard assessment and early-warning design, yet high-fidelity solvers remain computationally expensive, especially when large ensembles of scenarios must be explored under epistemic uncertainty in rheology, initial conditions, and topography. At the same time, field observations are typically sparse and heterogeneous, limiting the direct use of purely data-driven approaches. In this work, we develop a Deep-Learning Fourier Neural Operator (FNO) as a fast and accurate surrogate for one-dimensional shallow-water debris-flow simulations, and we demonstrate its application to characterizing the Rendinara–Morino debris-flow system in central Italy. A validated finite-volume solver with HLLC and Rusanov fluxes, Voellmy-type basal friction, hydrostatic reconstruction, and robust wet–dry treatment is used to generate a large ensemble of synthetic simulations over longitudinal profiles representative of the study area. The parameter space of bulk density, initial flow thickness, and Voellmy friction coefficients is systematically sampled, and the resulting space–time fields of flow depth and velocity form the training dataset. A two-dimensional FNO in the (x,t) plane is trained to map coordinates, rheological parameters, bed elevation, and initial conditions to the full evolution of depth and velocity. On a held-out validation set, the surrogate attains mean relative L2 errors below about 6% for flow depth and 10–15% for velocity, including prediction on an unseen topographic profile, while providing speed-ups of up to 36× (several orders of magnitude) compared to the numerical solver. These results show that combining physics-based synthetic data with operator-learning architectures enables the construction of site-specific, computationally efficient surrogates for debris-flow hazard analysis in data-scarce environments.
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