An RBF Approach for Enhanced Surrogate Modeling of a Debris Flow

In the last decades we have observed a rapid growth of extreme hydrological events, such as floods and rock/debris or mud flows affecting more and more frequently our lives. The detailed physical description of these viscous fluids is fundamental to understand the caused stress on possible flood control structures, such as levees, dams, check dams. However, its simulation through high fidelity physics-based computational models, using for example the Material Point Method (MPM), is extremely computationally demanding, thus limiting the application to real system monitoring.

The development of surrogate models to efficiently replicate the relevant features of the flow is of paramount importance to make a substantial step in the direction of real-time computations, required in any early warning system and to develop mitigation strategies.

Surrogate models have gained significant attention in recent years, especially with the advent of machine learning and the development of neural network-based methods, such as Fourier Neural Operators and Deep Operator Networks, among others. 
Here we consider surrogates based on Kernel methods, which demonstrated distinct advantages over widely used neural network-based approaches and provide rigorous error analysis. As fractal functions are pivotal in addressing nonlinear and irregular problems, we propose using the recently developed fractal RBFs as kernel of the surrogate model.

To demonstrate the effectiveness of the proposed approach, we consider a 2D debris flow along a 5m flume as a test scenario, where the outputs of interest are the position of the front and the velocities as functions of the fluid density and the inclination angle of the slope. 
Our results explore the accuracy and computational efficiency of the fractal RBF surrogate model compared to other kernel-based approaches.