IDENT: Identifying Differential Equations with Numerical Time Evolution

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 equatio...

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Vydané v:Journal of scientific computing Ročník 87; číslo 1; s. 1
Hlavní autori: Kang, Sung Ha, Liao, Wenjing, Liu, Yingjie
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.04.2021
Springer Nature B.V
Predmet:
Algorithms
Approximation
Boundary conditions
Computational Mathematics and Numerical Analysis
Deep learning
Dictionaries
Dimensional analysis
Dynamical systems
Error correction
Evolution
Identification
Incoherence
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Neural networks
Partial differential equations
Theoretical
Time dependence
Identifying unknown differential equations
Base element expansion (BEE)
Varying coefficients
Downsampling
Time evolution error (TEE)
Denoising
ISSN:0885-7474, 1573-7691
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Popis
Shrnutí: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.
Bibliografia:ObjectType-Article-1
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-020-01404-9