A review of shape-morphing solutions and evolutional neural networks for spatiotemporal dynamics

Shape-morphing solutions (SMS) refer to a class of approximate solutions of partial differential equations (PDEs) with the distinguishing feature that they depend nonlinearly on a set of time-dependent parameters. They generalize Galerkin truncations by allowing the basis (or trial functions) to evolve in time in order to adapt to the solution of the PDE. As such, SMS are particularly suitable for reduced-order modeling as well as high fidelity simulation of multiscale systems which exhibit localized time-dependent features, such as vortices, dispersive wave packets, and shocks. Furthermore, being mesh-free, SMS is scalable for solving PDEs in higher spatial dimensions. As a special case, SMS allows the approximation of the PDE's solution by a neural network whose weights and biases depend on time. Such neural networks are known as evolutional neural networks or neural Galerkin schemes. The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac-Frenkel variational principle. Over the past five years, contributions to the theory and computation of SMS have been growing rapidly. Here, we survey these developments, showcase some applications of SMS, and highlight important open problems for future research. At the same time, this review is structured to serve as a tutorial for applied mathematicians, physicist, and engineers who wish to enter this field.