The fractional Laplacian in R d, which we write as (− Δ) α/2 with α∈(0, 2), has multiple
equivalent characterizations. Moreover, in bounded domains, boundary conditions must be …

Computational modeling of the initiation and propagation of complex fracture is central to the
discipline of engineering fracture mechanics. This review focuses on two promising …

This handbook covers the peridynamic modeling of failure and damage. Peridynamics is a
reformulation of continuum mechanics based on integration of interactions rather than …

Fractional calculus, which has two main features—singularity and nonlocality from its origin—
means integration and differentiation of any positive real order or even complex order. It has …

The ubiquity of semilinear parabolic equations is clear from their numerous applications
ranging from physics and biology to materials and social sciences. In this paper, we …

Partial differential equations (PDEs) are used with huge success to model phenomena
across all scientific and engineering disciplines. However, across an equally wide swath …

The nonlocal Allen--Cahn equation, a generalization of the classic Allen--Cahn equation by
replacing the Laplacian with a parameterized nonlocal diffusion operator, satisfies the …

Physics-informed neural networks (PINNs) are effective in solving inverse problems based
on differential and integro-differential equations with sparse, noisy, unstructured, and multi …

A vector calculus for nonlocal operators is developed, including the definition of nonlocal
divergence, gradient, and curl operators and the derivation of the corresponding adjoint …

Abstract Neural operators [1],[2],[3],[4],[5] have recently become popular tools for designing
solution maps between function spaces in the form of neural networks. Differently from …