An optimization-based approach to parameter learning for fractional type nonlocal models

TL;DR

This paper introduces an optimization-based method for identifying parameters in fractional nonlocal models, improving the accuracy of simulations where classical diffusive assumptions do not hold.

Contribution

It formulates the parameter learning as an optimal control problem and proposes a gradient-based numerical approach with enhanced bilinear form approximations.

Findings

Method effectively estimates fractional order and truncation radius.

Numerical tests demonstrate robustness and applicability.

Approach improves parameter inference in nonlocal models.

Abstract

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.