應(yīng)用自適應(yīng)節(jié)點(diǎn)生成物理信息網(wǎng)絡(luò)計(jì)算地震波旅行時(shí)

摘要：求解程函方程能夠獲得震源定位、層析成像等地球物理反演所需的地震波旅行時(shí)，常用算法包括快速推進(jìn)法（FMM）和快速掃描法（FSM）等。物理信息神經(jīng)網(wǎng)絡(luò)（PINN）是一種新穎的無(wú)網(wǎng)格方法，可將偏微分方程中的微分形式約束條件融入到神經(jīng)網(wǎng)絡(luò)的損失函數(shù)中，從而獲得帶物理信息約束的神經(jīng)網(wǎng)絡(luò)。文中聚焦訓(xùn)練過(guò)程中的節(jié)點(diǎn)優(yōu)化配置，采用基于殘差分布的自適應(yīng)采樣方法改善PINN的訓(xùn)練效果，提出了基于自適應(yīng)節(jié)點(diǎn)生成的物理信息網(wǎng)絡(luò)旅行時(shí)計(jì)算方法。Marmousi模型和起伏地表模型的測(cè)試結(jié)果均表明，該方法相較于固定節(jié)點(diǎn)生成方法具有更穩(wěn)定的訓(xùn)練過(guò)程并且旅行時(shí)計(jì)算結(jié)果能保持較高的精度。

關(guān)鍵詞：旅行時(shí)計(jì)算，程函方程，物理信息神經(jīng)網(wǎng)絡(luò)（PINN），殘差，自適應(yīng)節(jié)點(diǎn)生成，無(wú)網(wǎng)格方法中圖分類號(hào)：P631 文獻(xiàn)標(biāo)識(shí)碼：A DOI：10.13810/j.cnki.issn.1000-7210.20240086

Traveltime calculation method with physics-informed neural network byresidual-based adaptive node generation

TANG Jie12， WANG Haicheng12，F(xiàn)AN Zhonghao12，PAN Deng12，REN Limin3，ZHANG Jingdong （1.SchoolofGeosciencesChina UniversityofPetroleu（EastChia）Qingdao，hndong2658O，Chna;2.Laboratoryforie MineralResourcesQingdaoNationalLaboratoryforMarineScienceand TechnologyQingdao，Shandong 266071，China ;3.BGP Equipment，CNC，Zhuozhou，HebeiO751，China;4.GeologicalResearchCenterGPIc.，ZhuozhouCNPC，HebeiOCina）

Abstract： Seismic wave traveltime information required for geophysical inversion such as source localization and tomographic imaging can be obtained by solving the eikonal：equation. Common algorithms for this purpose include the fast marching method （FMM） and the fast sweeping method （FSM）.Physics-informed neural network （PINN） is a novel mesh-free method that incorporates the constraints of partial differential equations into the loss function of the neural network，thus embedding physical information into the network. Focusing on optimizing node distribution during training，this study adopts an adaptive sampling strategy based on residual distributio to improve the training performance of PINN and proposes a travel-time calculation method using PINN with adaptive node generation. Application tests on the Marmousi model and an iregular topography model show that， compared with the fixed node-generation method，the proposed approach yields a more stable training process and maintains high accuracy in traveltime calculations.

Keywords：traveltime calculation，eikonal equation，physics-informed neural network（PINN），residual，adaptive nodegeneration，mesh-free method

唐杰，王海成，范忠豪，等，應(yīng)用自適應(yīng)節(jié)點(diǎn)生成物理信息網(wǎng)絡(luò)計(jì)算地震波旅行時(shí)[J].石油地球物理勘探， 2025，60（4） ：840-851. TANG Jie， WANG Haicheng，F(xiàn)AN Zhonghao，et al.Traveltime calculation method with physics-informed neural network by residual-based adaptive node generation[J]. Oil Geophysical Prospecting，2025，60（4）：840-851.

0 引言

波初至旅行時(shí)，是地震資料處理和成像所需的重要信息[1-3]。傳統(tǒng)的程函方程求解主要有兩種常用算法，分別是快速推進(jìn)法（FastMarchingMethod，F(xiàn)MM）和快速掃描法（FastSweepingMethod，程函方程（EikonalEquation）常被用來(lái)計(jì)算地震

FSM）。Sethian4采用迎風(fēng)差分格式離散程函方程，再通過(guò)FMM計(jì)算旅行時(shí)分布。FSM將旅行時(shí)場(chǎng)傳播方向分成多組，對(duì)每組分別利用Gauss-Seidel迭代方法求解離散化后的方程[5-6]。這兩種方法都能夠計(jì)算初至旅行時(shí)，但是對(duì)于結(jié)構(gòu)復(fù)雜模型，在精度和計(jì)算效率方面仍有提升空間。

近年來(lái)，深度學(xué)習(xí)方法得到了廣泛的研究及應(yīng)用。……