Optimization-based schemes to solve PDEs with nonlocal conditions

Partial differential equations (PDEs) with nonlocal initial or boundary conditions arise in a wide range of scientific and engineering applications, including heat conduction with integral constraints, population dynamics, anomalous diffusion, materials with memory, and models of fluid flow and transport influenced by global conservation laws. These nonlocal formulations often provide a more accurate description of physical processes than classical local conditions, but they also introduce additional computational challenges.

In recent years, Physics-Informed Neural Networks (PINNs) have been proposed as a flexible framework for solving PDEs without explicit discretization of the domain. Despite their growing popularity, the computational efficiency of PINNs for PDE problems, especially those involving nonlocal constraints, remains insufficiently explored. This work investigates optimization-based approaches for solving parabolic and elliptic PDEs with nonlocal conditions.

Two numerical approaches are investigated. The first method employs finite difference discretization of the PDEs, from which a discrete residual-based loss function is constructed. This loss is minimized using three distinct optimization algorithms: ADAM, AMSGrad and L-BFGS. The second method uses PINNs, where the PDE operators and nonlocal constraints are embedded directly into the training loss. The same three optimizers and stopping criteria are applied to both approaches to ensure a consistent comparison, and in both cases gradients are computed using automatic differentiation.

The methods are evaluated on representative parabolic and elliptic PDE test problems with nonlocal conditions in regular domains. Performance is assessed primarily in terms of execution time required to achieve a given accuracy. The comparison is extended by fixing the spatial domain and increasing the density of discretization points. This allows for a consistent evaluation of how execution time scales with spatial resolution for both approaches.