Boundary integrated neural networks and code for acoustic radiation and scattering

Abstract

This paper presents a novel approach called the boundary integrated neural networks (BINNs) for analyzing acoustic radiation and scattering. The method introduces fundamental solutions of the time-harmonic wave equation to encode the boundary integral equations (BIEs) within the neural networks, replacing the conventional use of the governing equation in physics-informed neural networks (PINNs). This approach offers several advantages. First, the input data for the neural networks in the BINNs only require the coordinates of “boundary” collocation points, making it highly suitable for analyzing acoustic fields in unbounded domains. Second, the loss function of the BINNs is not a composite form and has a fast convergence. Third, the BINNs achieve comparable precision to the PINNs using fewer collocation points and hidden layers/neurons. Finally, the semianalytic characteristic of the BIEs contributes to the higher precision of the BINNs. Numerical examples are presented to demonstrate the performance of the proposed method, and a MATLAB code implementation is provided as supplementary material.

Keywords

acoustic / semianalytical / physics-informed neural networks / boundary integral equations / boundary integral neural networks / unbounded domain

References

[1] Koo BU. Shape design sensitivity analysis of acoustic problems using a boundary element method. Comput Struct. 1997;65(5):713-719.

[2] Zheng C, Matsumoto T, Takahashi T, Chen H. A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method. Eng Anal Bound Elem. 2012;36(3):361-371.

[3] Chen L, Zheng C, Chen H. A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method. Comput Mech. 2013;52:631-648.

[4] Jordan MI, Mitchell TM. Machine learning: trends, perspectives, and prospects. Science. 2015;349(6245):255-260.

[5] Ruthotto L, Haber E. Deep neural networks motivated by partial differential equations. J Math Imaging Vision. 2020;62:352-364.

[6] Zeng S, Zhang Z, Zou Q. Adaptive deep neural networks methods for high-dimensional partial differential equations. J Comput Phys. 2022;463:111232.

[7] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686-707.

[8] Yang L, Meng X, Karniadakis GE. B-PINNs: Bayesian physicsinformed neural networks for forward and inverse PDE problems with noisy data. J Comput Phys. 2021;425:109913.

[9] Gu Y, Zhang C, Zhang P, Golub MV, Yu B. Enriched physics-informed neural networks for 2D in-plane crack analysis: theory and MATLAB code. Int J Solids Struct. 2023;276:112321.

[10] Sirignano J, Spiliopoulos K. DGM: a deep learning algorithm for solving partial differential equations. J Comput Phys. 2018;375:1339-1364.

[11] Saporito YF, Zhang Z. Path-dependent deep Galerkin method: a neural network approach to solve path-dependent partial differential equations. SIAM J Financial Math. 2021;12(3):912-940.

[12] E W, Yu B. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat. 2018;6(1):1-12.

[13] Gu Y, Ng MK. Deep Ritz method for the spectral fractional laplacian equation using the Caffarelli–Silvestre extension. SIAM J Scientific Comput. 2022;44(4):A2018-A2036.

[14] Song C, Alkhalifah T, Waheed UB. Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks. Geophys J Int. 2021;225(2):846-859.

[15] Zhang Y, Zhu X, Gao J. Seismic inversion based on acoustic wave equations using physics-informed neural network. IEEE Trans Geosci Remote Sensing. 2023;61:1-11.

[16] Lin G, Hu P, Chen F, et al. BINet: learn to solve partial differential equations with boundary integral networks. CSIAM Trans Appl Math. 2023;4:275-305.

[17] Zhang H, Anitescu C, Bordas S, Rabczuk T, Atroshchenko E. Artificial neural network methods for boundary integral equations. ???TechRxiv Preprint techrxiv20164769v1???. 2022.

[18] Sun J, Liu Y, Wang Y, Yao Z, Zheng X. BINN: a deep learning approach for computational mechanics problems based on boundary integral equations. Comput Methods Appl Mech Eng. 2023;410:116012.

[19] Qu W, Zheng C, Zhang Y, Gu Y, Wang F. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct. 2018;206:82-89.

[20] Schot SH. Eighty years of Sommerfeld’s radiation condition. Historia Math. 1992;19:385-401.

[21] Qu W, Chen W, Zheng C. Diagonal form fast multipole singular boundary method applied to the solution of high-frequency acoustic radiation and scattering. Int J Numer Meth Eng. 2017;111:803-815.

[22] Guiggiani M, Casalini P. Direct computation of Cauchy principal value integrals in advanced boundary elements. Int J Numer Meth Eng. 1987;24(9):1711-1720.

[23] Zhang Y. Evaluation of singular and nearly singular integrals in the BEM with exact geometrical representation. J Computat Math. 2013;31:355-369.

[24] Xie G, Zhou F, Li H, Wen X, Meng F. A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method. Front Mech Eng. 2019;14:332-341.

[25] Xie G, Zhang D, Meng F, Du W, Zhang J. Calculation of stress intensity factor along the 3D crack front by dual BIE with new crack front elements. Acta Mech. 2017;228:3135-3153.

[26] Qu W, Chen W, Fu Z. Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method. Eng Anal Bound Elem. 2015;60:2-9.

[27] Qu W, Gu Y, Fan CM. A stable numerical framework for long-time dynamic crack analysis. Int J Solids Struct. 2024;293:112768.

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2024 The Authors. International Journal of Mechanical System Dynamics published by John Wiley & Sons Australia, Ltd on behalf of Nanjing University of Science and Technology.