Sequential data assimilation for PDEs using shape-morphing solutions

TL;DR

This paper introduces a sequential data assimilation method for shape-morphing solutions of PDEs, combining predictor-corrector schemes with Newton-like iterations to improve solution accuracy using observational data.

Contribution

The paper presents a novel data assimilation approach for shape-morphing solutions of PDEs, demonstrating convergence and effectiveness with sparse data and minimal iterations.

Findings

DA-SMS converges uniformly to the true system state.

Effective with sparse observations and a single Newton iteration.

Validated on nonlinear Schrödinger, Kuramoto-Sivashinsky, and advection-diffusion equations.

Abstract

Shape-morphing solutions (also known as evolutional deep neural networks, reduced-order nonlinear solutions, and neural Galerkin schemes) are a new class of methods for approximating the solution of time-dependent partial differential equations (PDEs). Here, we introduce a sequential data assimilation method for incorporating observational data in a shape-morphing solution (SMS). Our method takes the form of a predictor-corrector scheme, where the observations are used to correct the SMS parameters using Newton-like iterations. Between observation points, the SMS equations (a set of ordinary differential equations) are used to evolve the solution forward in time. We prove that, under certain conditions, the data assimilated SMS (DA-SMS) converges uniformly towards the true state of the system. We demonstrate the efficacy of DA-SMS on three examples: the nonlinear Schrodinger equation, the Kuramoto-Sivashinsky equation, and a two-dimensional advection-diffusion equation. Our numerical results suggest that DA-SMS converges with relatively sparse observations and a single iteration of the Newton-like method.