Exact solution of a delay difference equation modeling traffic flow and their ultra-discrete limit

Matsuya, Keisuke; Kanai, Masahiro

doi:10.48550/arxiv.1509.07861

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…where a, b, c are real constants. Here we include a free parameter α to the reduction condition such as (6). By applying the reduction, we obtain a discrete equation which depends on the free parameter α, the discrete variable n, m, and the time-lattice parameter δ which is defined by the parameters a, b, c such as (7).…”

Section: A Methods To Construct Delay-analogues Of Soliton Equationsmentioning

confidence: 99%

“…where f m n ≡ f n,m,k . More precisely, applying the reduction condition (6) to equation (1), we have made the index k + 1 of f become k. Then we can omit the variable k since the iterations of k vanish. We can rewrite the bilinear equation (7) as follows by using Hirota's D-operators:…”

Section: An Integrable Delay Lotka-volterra Equationmentioning

confidence: 99%

“…Because of their importance in various fields, the mathematical properties of delay differential equations have been actively studied. For example, exact solutions of delay differential equations play an important role in the study of traffic flow [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

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A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

Nakata¹,

Maruno²

2022

J. Phys. A: Math. Theor.

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We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka–Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter approaches 0.

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Section: A Methods To Construct Delay-analogues Of Soliton Equationsmentioning

confidence: 99%

Section: An Integrable Delay Lotka-volterra Equationmentioning

confidence: 99%

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A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

Nakata¹,

Maruno²

2022

J. Phys. A: Math. Theor.

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“…The non-Markovian closure terms with time-lagged state information lead us to the use of delay differential equations (DDEs) [23]. DDEs have been widely used in many fields such as biology [84,54], pharmacokineticpharmacodynamics [39], chemistry [78], economics [38], transportation [55], control theory [43], climate dynamics [30,6], etc. Next, we summarize the state of the art for learning and solving differential equations using neural-networks (NNs) and develop theory and schemes for neural DDEs including adjoint equations for backpropagation.…”

Section: Neural Delay Differential Equationsmentioning

confidence: 99%

Neural Closure Models for Dynamical Systems

Gupta,

Lermusiaux

2020

Preprint

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Complex dynamical systems are used for predictions in many applications. Because of computational costs, models are however often truncated, coarsened, or aggregated. As the neglected and unresolved terms along with their interactions with the resolved ones become important, the usefulness of model predictions diminishes. We develop a novel, versatile, and rigorous methodology to learn non-Markovian closure parameterizations for low-fidelity models using data from high-fidelity simulations. The new neural closure models augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori-Zwanzig formulation and the inherent delays in natural dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgridscale processes in coarse models, and augment the simplification of complex biochemical models. We show that using non-Markovian over Markovian closures improves long-term accuracy and requires smaller networks. We provide adjoint equation derivations and network architectures needed to efficiently implement the new discrete and distributed nDDEs. The performance of discrete over distributed delays in closure models is explained using information theory, and we observe an optimal amount of past information for a specified architecture. Finally, we analyze computational complexity and explain the limited additional cost due to neural closure models.

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