[논문]물리정보신경망을 이용한 파동방정식 모델링 전략 분석

조상인; 최우창; 지준; 편석준

doi:10.7582/gge.2023.26.3.114

물리정보신경망을 이용한 파동방정식 모델링 전략 분석
Analysis on Strategies for Modeling the Wave Equation with Physics-Informed Neural Networks 원문보기

지구물리와 물리탐사 = Geophysics and geophysical exploration, v.26 no.3, 2023년, pp.114 - 125

조상인 (인하대학교 에너지자원공학과) , 최우창 (인하대학교 에너지자원공학과) , 지준 (한성대학교 AI응용학과) , 편석준 (인하대학교 에너지자원공학과)

초록
AI-Helper

편미분방정식의 해를 구하기 위한 여러 수치해법들의 한계와 순수 데이터 기반 기계학습의 단점을 극복하기 위해 물리정보신경망(physics-informed neural network, PINN)이 제안되었다. 물리정보신경망은 편미분방정식을 손실함수 구성에 직접 활용하여 기계학습 훈련에 물리적 제약을 주는 기법으로 파동방정식 모델링에도 활용될 수 있다. 그러나 물리정보신경망을 이용하여 파동방정식을 풀기 위해서는 신경망 훈련 시 입력에 대한 2차 미분이 수행되어야 하고, 그 결과로 출력되는 파동장은 복잡한 역학적 현상들을 포함하고 있어 섬세한 전략이 필요하다. 이 해설 논문에서는 물리정보신경망의 기본 개념을 설명하고 파동방정식 모델링에 활용하기 위한 고려사항들에 대해 고찰하였다. 이러한 고려사항에는 공간좌표 정규화, 활성함수 선정, 물리손실 추가 전략이 포함된다. 훈련자료의 공간좌표를 정규화한 후 사용하면 파동방정식 모델링을 위한 신경망 훈련에서 초기 조건이 더 정확하게 반영되는 것을 수치 실험을 통해 보였다. 또한 신경망을 통한 파동장 예측에 가장 적절한 활성함수를 선정하기 위해 여러 함수들의 특성을 비교했다. 특성 비교는 각 활성함수들의 입력자료에 대한 미분과 수렴성을 중심으로 이루어졌다. 마지막으로 신경망 훈련 중 손실함수에 물리손실을 추가하는 두가지 시나리오의 결과를 비교하였다. 수치 실험을 통해 훈련 초기부터 물리손실을 활용하는 전략보다 초기 훈련단계 이후부터 물리손실을 적용하는 커리큘럼 기반 학습전략이 효과적이라는 결과를 도출했다. 추가로 이 결과를 물리손실을 전혀 사용하지 않은 훈련 결과와 비교하여 PINN기법의 효과를 확인하였다.

Abstract ▼ AI-Helper

The physics-informed neural network (PINN) has been proposed to overcome the limitations of various numerical methods used to solve partial differential equations (PDEs) and the drawbacks of purely data-driven machine learning. The PINN directly applies PDEs to the construction of the loss function, introducing physical constraints to machine learning training. This technique can also be applied to wave equation modeling. However, to solve the wave equation using the PINN, second-order differentiations with respect to input data must be performed during neural network training, and the resulting wavefields contain complex dynamical phenomena, requiring careful strategies. This tutorial elucidates the fundamental concepts of the PINN and discusses considerations for wave equation modeling using the PINN approach. These considerations include spatial coordinate normalization, the selection of activation functions, and strategies for incorporating physics loss. Our experimental results demonstrated that normalizing the spatial coordinates of the training data leads to a more accurate reflection of initial conditions in neural network training for wave equation modeling. Furthermore, the characteristics of various functions were compared to select an appropriate activation function for wavefield prediction using neural networks. These comparisons focused on their differentiation with respect to input data and their convergence properties. Finally, the results of two scenarios for incorporating physics loss into the loss function during neural network training were compared. Through numerical experiments, a curriculum-based learning strategy, applying physics loss after the initial training steps, was more effective than utilizing physics loss from the early training steps. In addition, the effectiveness of the PINN technique was confirmed by comparing these results with those of training without any use of physics loss.

주제어

표/그림 (9)

그림 Fig. 1. Schematic diagram of PINN to solve the forward problem of the 2D acoustic wave equation.
표 Table 1. Definition of the activation functions and their 1^st and 2^nd derivative.
그림 Fig. 2. Comparison between (a) four activation functions (ReLU, Tanh, Softplus and Siren), their (b) 1^st derivative, and (c) 2^nd derivative.
그림 Fig. 3. Structure of the neural network used to solve the wave equation (modified after Moseley et al. (2020b)).
그림 Fig. 4. Wavefield prediction from neural network training (a) without coordinate normalization and with normalization that results in coordinate ranging (b) -0.5 ~ 0.5, and (c) -1 ~ 1.
그림 Fig. 5. Comparison of loss functions between four activation functions (ReLU, Tanh, Softplus, and Siren).
그림 Fig. 6. Comparison between (a) label data generated with the FDM method and the results of the wavefield prediction at 0.04 s by neural network training with (b) ReLU, (c) Softplus, (d) Tanh, and (e) Siren activation functions.
그림 Fig. 7. Comparison between (a) label data generated with the FDM and the results of the wavefield prediction by neural network training (b) with and (d) without the curriculum-based learning strategy. (c) and (e) show the differences between (a) and (b), and (a) and (d), respectively.
그림 Fig. 8. (a) Results of the wavefield prediction achieved through neural network training without physics loss and (b) the differences observed compared to the wavefield generated by the FDM (Fig. 8a).

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