Nicolás Guarín-Zapata

Gregorio Pérez-Bernal, Oscar Rincón-Cardeño, Silvana Montoya-Noguera et al.

Inverse problems are extensively studied in applied mathematics, with applications ranging from acoustic tomography for medical diagnosis to geophysical exploration. Physics informed neural networks (PINNs) have emerged as a powerful tool for solving such problems, while Physics informed Kolmogorov Arnold networks (PIKANs) represent a recent benchmark that, in certain problems, promises greater interpretability and accuracy compared to PINNs, due to their nature, being constructed as a composition of polynomials. In this work, we develop a methodology for addressing inverse problems in infinite and semi infinite domains. We introduce a novel sampling strategy for the network's training points, using the negative exponential and normal distributions, alongside a dual network architecture that is trained to learn the solution and parameters of an equation with the same loss function. This design enables the solution of inverse problems without explicitly imposing boundary conditions, as long as the solutions tend to stabilize when leaving the domain of interest. The proposed architecture is implemented using both PINNs and PIKANs, and their performance is compared in terms of accuracy with respect to a known solution as well as computational time and response to a noisy environment. Our results demonstrate that, in this setting, PINNs provide a more accurate and computationally efficient solution, solving the inverse problem 1,000 times faster and in the same order of magnitude, yet with a lower relative error than PIKANs.

Oscar Rincón-Cardeno, Gregorio Pérez Bernal, Silvana Montoya Noguera et al.

Purpose - This study compares the Boundary Element Method (BEM) and Physics-Informed Neural Networks (PINNs) for solving the two-dimensional Helmholtz equation in wave scattering problems. The objective is to evaluate the performance of both methods under the same conditions. Design/methodology/approach - We solve the Helmholtz equation using BEM and PINNs for the same scattering problem. The PINNs are trained by minimizing the residual of the governing equations and boundary conditions, with their configuration determined through hyperparameter optimization, while the BEM is applied using boundary discretization. Both methods are evaluated in terms of solution accuracy, computation time, and generalization capacity. Findings - Numerical experiments were conducted by varying the number of integration points for BEM and the number of layers and neurons per layer for PINNs. Hyperparameter tuning provided further insight into suitable configurations for wave scattering problems. At comparable accuracy, PINNs produced consistent solutions but required training times approximately 42 times longer than BEM. However, once trained, PINNs achieved evaluation times up to 204 times faster. The generalization capacity was also assessed outside the PINN training domain, where the relative error increased from $7.46 \times 10^{-2}$ to 8.22, while BEM maintained a similar error level in the extended region. Originality/value - This work presents a direct comparison between PINNs and BEM for the Helmholtz equation. The analysis provides quantitative data on the performance of both methods, supporting their selection in future research on wave propagation problems and establishing future challenges and directions.

Juan Fernando Riascos-Goyes, Michael Lowry, Nicolás Guarín-Zapata et al.

Urban morphology has long been recognized as a factor shaping human mobility, yet comparative and formal classifications of urban form across metropolitan areas remain limited. Building on theoretical principles of urban structure and advances in unsupervised learning, we systematically classified the built environment of nine U.S. metropolitan areas using structural indicators such as density, connectivity, and spatial configuration. The resulting morphological types were linked to mobility patterns through descriptive statistics, marginal effects estimation, and post hoc statistical testing. Here we show that distinct urban forms are systematically associated with different mobility behaviors, such as reticular morphologies being linked to significantly higher public transport use (marginal effect = 0.49) and reduced car dependence (-0.41), while organic forms are associated with increased car usage (0.44), and substantial declines in public transport (-0.47) and active mobility (-0.30). These effects are statistically robust (p < 1e-19), highlighting that the spatial configuration of urban areas plays a fundamental role in shaping transportation choices. Our findings extend previous work by offering a reproducible framework for classifying urban form and demonstrate the added value of morphological analysis in comparative urban research. These results suggest that urban form should be treated as a key variable in mobility planning and provide empirical support for incorporating spatial typologies into sustainable urban policy design.