Discovery of Dynamics Using Linear Multistep Methods

Keller, Rachael T.; Du, Qiang

doi:10.1137/19m130981x

Cited by 29 publications

(28 citation statements)

References 31 publications

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“…The discovery of subdiffusion is essentially an inverse process of solving a subdiffusion problem (1.2). That means, suppose that only the information of u(x m ,t n ) at the pairs of the uniform grid points {x m ,t n } is provided, we need to recover the source function f [14]. Assume u(x,t) and f (x,t) both are unknown in subfiffuion system (1.1) with given u n m = u(x m ,t n ), the target is to approximate the close-form expression for f (x m ,t n ).…”

Section: Discovery Of Subdiffusionmentioning

confidence: 99%

“…Consider the neural network approximation via L 1 discertization. We use N as the set of all neural networks with special architecture [14]. Now we introduce a network f (•) ∈ N to approximate f (•).…”

Section: Discovery Of Subdiffusion In Deep Learningmentioning

confidence: 99%

“…Based on the the neural networks, the data-driven discovery of differential equations has been proposed in [29,23,25,22,21,7] including the space-fractional differential equations [12]. In particular, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in [7,14] and [29]. Identifying source function from observed data is a meaningful and challenge topic for the time-dependent PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Discovery of subdiffusion problem with noisy data via deep learning

Xu¹,

Chen²

2022

Preprint

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Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the discovery problem. Recently, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in [Racheal and Du, SIAM J. Numer. Anal. 59 (2021) 429-455; Du et al. arXiv:2103.11488]. We extend this framework to the data-driven discovery of the timefractional PDEs, which can effectively characterize the ubiquitous power-law phenomena. In this paper, identifying source function of subdiffusion with noisy data using L 1 approximation in deep neural network is presented. In particular, two types of networks for improving the generalization of the subdiffusion problem are designed with noisy data. The numerical experiments are given to illustrate the availability using deep learning. To the best of our knowledge, this is the first topic on the discovery of subdiffusion in deep learning with noisy data.

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Section: Discovery Of Subdiffusionmentioning

confidence: 99%

Section: Discovery Of Subdiffusion In Deep Learningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discovery of subdiffusion problem with noisy data via deep learning

Xu¹,

Chen²

2022

Preprint

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“…In solving dynamical systems, high-order discretization techniques such as linear multistep methods (LMMs) and Runge-Kutta methods have been well-developed [4,13,30]. Recently, LMMs have been employed for the discovery of dynamics [41,50,54,19]. In [19], a rigorous framework based on refined notions of consistency and stability is established to yield the convergence of LMM-based discovery for three popular LMM schemes (the Adams-Bashforth, Adams-Moulton, and Backwards Differentiation Formula schemes).…”

mentioning

confidence: 99%

“…Recently, LMMs have been employed for the discovery of dynamics [41,50,54,19]. In [19], a rigorous framework based on refined notions of consistency and stability is established to yield the convergence of LMM-based discovery for three popular LMM schemes (the Adams-Bashforth, Adams-Moulton, and Backwards Differentiation Formula schemes). However, the theory in [19] is specialized for methods that cannot provide a closed-form expression for the governing function, which is needed in many applications.…”

mentioning

confidence: 99%

The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation

Du¹,

Gu²,

Yang³

et al. 2021

Preprint

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Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development.In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the ℓ 2 grid error is bounded by the sum of O(h p ) and the network approximation error, where h is the time step size and p is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.

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