Abstract

Identifying unknown differential equations from a given set of discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable. Nonlinearity and varying coefficients add complexity to the problem. We assume that the governing partial differential equation (PDE) is a linear combination of few differential terms in a prescribed dictionary, and the objective of this paper is to find the correct coefficients. We propose a new direction based on the fundamental convergence principle of numerical PDE schemes. We utilize Lasso for efficiency, and a performance guarantee is established based on an incoherence property. The main contribution is to validate and correct the results by time evolution error (TEE). A new algorithm, called identifying differential equations with numerical time evolution (IDENT), is explored for data with non-periodic boundary conditions, noisy data and PDEs with varying coefficients. Based on the recovery theory of Lasso, we propose a new definition of Noise-to-Signal ratio, which better represents the level of noise in the case of PDE identification. The effects of data generations and downsampling are systematically analyzed and tested. For noisy data, we propose an order preserving denoising method called least-squares moving average (LSMA), to preprocess the given data. For the identification of PDEs with varying coefficients, we propose to add Base Element Expansion (BEE) to aid the computation. Various numerical experiments from basic tests to noisy data, downsampling effects and varying coefficients are presented.

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.