This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

DeepONets have recently been proposed as a framework for learning nonlinear operators
mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove …

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 …

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 …

The energy dissipation law and the maximum bound principle (MBP) are two important
physical features of the well-known Allen--Cahn equation. While some commonly used first …

In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the
maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper …

The maximum bound principle (MBP) is an important property for a large class of semilinear
parabolic equations, in the sense that the time-dependent solution of the equation with …

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation
which contains small perturbation parameters and strong nonlinearity. A standard fully …

Gradient flow models attract much attention these years. The energy naturally decreases
along the direction of gradient flows. In order to preserve this property, various numerical …

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP)
in the sense that the time-dependent solution preserves for any time a uniform pointwise …