Methods to Recover Unknown Processes in Partial Differential Equations Using Data

Chen, Zhe; Wu, Kailiang; Xiu, Dongbin

doi:10.1007/s10915-020-01324-8

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…we input b and W computed through (9), with subsampling in (11). For each sparsity level k = 1, 2, .…”

Section: Algorithmsmentioning

confidence: 99%

“…To recover the coefficient values using the support A k j , we find the row index set H of highly dynamic regions in (24), and solve W narrow c = bnarrow in (26) and get c(k, j) in (27). (8) uniformly subsampled as (11); Parameter T = 0.05 [12] and set j = 0; [Step 2] Find c(k, j) by narrow-fit (26);…”

Section: Algorithmsmentioning

confidence: 99%

“…In Figure 15, we show the effects of changing the final time T (the top row), and of changing ∆x and ∆t for N x N t (the second row), and of changing the uniform subsampling in (11), i.e., ∆t * and ∆x * for the generation of the feature matrix (the third row). We compare for the KS equation (34), the 2D linear ODE system (39), the Van der Pol equation (40), and the Duffing equation (41) to illustrate the effects.…”

Section: Effects Of Subsampling In Data Acquisition and The Feature M...mentioning

confidence: 99%

“…Differential equations with high-order derivatives, including the Korteweg-De Vries (KdV) equation, the Kuramoto-Sivashinsky (KS) equation, and 2D reaction-diffusion equations can be recovered even with a significant amount of noise. A related work [11] focuses on the identification of advection-diffusion equations, and shows that a Galerkin-type algorithm using the weak form out-performs the collocation-type algorithm based on using a differential form.…”

Section: Introductionmentioning

confidence: 99%

“…Figure15: Effects of the final time T (the top row), ∆t, ∆x for N x N t of the given data (the second row), and subsampling ∆t * , ∆x * in(11) in the third row. Each graph shows the average of the E 2 error, the TPR and PPV values over 20 experiments for one varying variable among the variables in {T, ∆x, ∆t, ∆x * , ∆t * } while the rest is fixed.…”

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confidence: 99%

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WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming

Tang¹,

Liao²,

Kuske³

et al. 2022

Preprint

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Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise. For a robust recovery against noise and the choice of hyper-parameters, we introduce two new mechanisms, narrow-fit and trimming, for the coefficient support and value recovery, respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming of the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to 100% noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.

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“…we input b and W computed through (9), with subsampling in (11). For each sparsity level k = 1, 2, .…”

Section: Algorithmsmentioning

confidence: 99%