Physics-informed neural networks to solve inverse problems in unbounded domains

Abstract: 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.