基于离散物理信息神经网络的线性偏微分方程长时积分

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关键词：物理信息神经网络;长时积分;时间离散;线性偏微分方程;热启动;双精度计算;波动方程;对流方程中图分类号：TP183 文献标志码：Adoi：10.12415/j.issn.1671-7872.24157

Abstract： An improved discrete physics-informed neural network method incorporating both warm-start techniques and double-precision computing was proposed to address the temporal causality failure encountered in continuous physics-informed neural networks for long-time integration of partial differential equations.The temporal causality compliance of neural network was enhanced through the introduction of a time-discretization strategy，while computational resourceconsumption was reduced by employing a warm-start scheme that utilized model parameters obtained from previous time-step training as initial parameters for subsequent time steps.In terms ofcomputational accuracy， the solution precision for linear partial diferential equations Was improved through the implementation of double-precision floating-point arithmetic.To validate the efectiveness of the proposed method，comparative numerical experiments were designed involving both convection equations and wave equations. The results demonstrate that the application of warm-start techniques achieves a 2-3 fold improvement in computational efficiency while providing approximately 20% enhancement in computational accuracy. For long-time integration tasks，the implementation of double-precision computing is shown to yield about 10 times higher computational accuracy， though accompanied bya7-8 fold increase in computation time.Furthermore，when additional physical constraintsare incorporated，the computational accuracy is observed to be further improved by5-6 times without significant additionalcomputational overhead.Through thecombinedapplicationof warm-start schemes and doubleprecisioncomputing， high-accuracy long-time integration is succesfully realized by the discrete physics-informed neural networks within controllble computational costs，thereby ofering a novel solution for prolonged simulations of complex systems.

Keywords：physics-informed neural network;long-time integration; time-discretization; partial differentialequation; warm-start; double precision calculation; wave equation; convection equation

偏微分方程 （partialdifferential equation，PDE）在科学和工程领域中[1-2]被广泛应用于各类现象的建模与分析。（剩余12786字）