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
map** 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 …

We propose a very general framework for deriving rigorous bounds on the approximation
error for physics-informed neural networks (PINNs) and operator learning architectures such …

Although very successfully used in conventional machine learning, convolution based
neural network architectures--believed to be inconsistent in function space--have been …

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 …

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 …

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn
equation has been an open problem in recent years. This work provides a positive answer …

We propose and analyze a class of temporal up to fourth-order unconditionally structure-
preserving single-step methods for Allen–Cahn-type semilinear parabolic equations. We first …

We present a systematic two-step approach to derive temporal up to the eighth-order,
unconditionally maximum-principle-preserving schemes for a semilinear parabolic sine …