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Sequential data assimilation for PDEs using shape-morphing solutions
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DOI：10.1016/j.jcp.2025.113994.png)
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Abstract
En 中文
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 DASMS on three examples: the nonlinear Schr & ouml;dinger 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.
Keywords:
Partial differential equations
Data assimilation
Machine learning
Dynamical systems
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