Solução da equação eikonal com redes neurais informadas por física e redes de Kolmogorov-Arnold

Abstract

This work presents an innovative approach for solving the Eikonal Equation, fun- damental for seismic modeling, using a hybrid neural network that combines Physics- Informed Neural Networks (PINNs) and Kolmogorov-Arnold Networks (KANs). The central objective was to develop a model capable of learning the solution of the partial differential equation by incorporating the laws of physics directly into its loss function. The data for training and validation were numerically generated using the PyKonal li- brary for a heterogeneous two-dimensional velocity model. The methodology evaluated two types of datasets: Type I, with concentrated seismic sources in a specific region, and Type II, with sources distributed throughout the entire grid domain to test the model’s generalization capability. The results for the Type I dataset demonstrated high accuracy, with a coefficient of determination of 0.9934, but failed to generalize to sources outside the training area. In contrast, the model trained with the Type II dataset mitigated the ge- neralization problem, obtaining consistent predictions across the entire grid. The analysis revealed that the spatial distribution of the samples is a crucial factor for performance, at times more impactful than the sampling density. The study suggests that the PINN-KAN architecture is a viable and promising alternative for the solution of the Eikonal Equation, with the caveat that the training data sampling strategy is fundamental to ensuring the generalization and accuracy of the solution.