2次元静磁場問題のNon-overlapping型領域分割法に基づくPINN
PINNは偏微分方程式の初期境界値問題の誤差を損失関数に組み込んでニューラルネットワークを学習させる手法であり，多くの研究が報告されている．PINNによる予測解の精度を向上させるためには，学習データセットのサイズを大きくし，さらに非一様度の低い分布を持つ学習データを用いることが望ましい．しかし，大規模な学習データを並列処理する場合，分布の特徴を維持したまま点集合を分割することは困難である．そこで本論文では，有限要素法の並列数値計算法として知られているNon-overlapping型領域分割法に着目する．特に，古典的なDirichlet-Neumann, Neumann-Neumann, Dir...
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| Published in | 日本計算工学会論文集 Vol. 2024; p. 20240009 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | Japanese |
| Published |
一般社団法人 日本計算工学会
23.08.2024
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1347-8826 |
| DOI | 10.11421/jsces.2024.20240009 |
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| Abstract | PINNは偏微分方程式の初期境界値問題の誤差を損失関数に組み込んでニューラルネットワークを学習させる手法であり，多くの研究が報告されている．PINNによる予測解の精度を向上させるためには，学習データセットのサイズを大きくし，さらに非一様度の低い分布を持つ学習データを用いることが望ましい．しかし，大規模な学習データを並列処理する場合，分布の特徴を維持したまま点集合を分割することは困難である．そこで本論文では，有限要素法の並列数値計算法として知られているNon-overlapping型領域分割法に着目する．特に，古典的なDirichlet-Neumann, Neumann-Neumann, Dirichlet-Dirichletアルゴリズムに加え，偏微分方程式として記述した共役勾配法に基づくDDMアルゴリズムを導出し，PINNを適用した．さらに，提案手法を2次元静磁場問題に適用し，数値実験結果により有用性を示した． |
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| AbstractList | PINNは偏微分方程式の初期境界値問題の誤差を損失関数に組み込んでニューラルネットワークを学習させる手法であり，多くの研究が報告されている．PINNによる予測解の精度を向上させるためには，学習データセットのサイズを大きくし，さらに非一様度の低い分布を持つ学習データを用いることが望ましい．しかし，大規模な学習データを並列処理する場合，分布の特徴を維持したまま点集合を分割することは困難である．そこで本論文では，有限要素法の並列数値計算法として知られているNon-overlapping型領域分割法に着目する．特に，古典的なDirichlet-Neumann, Neumann-Neumann, Dirichlet-Dirichletアルゴリズムに加え，偏微分方程式として記述した共役勾配法に基づくDDMアルゴリズムを導出し，PINNを適用した．さらに，提案手法を2次元静磁場問題に適用し，数値実験結果により有用性を示した． |
| Author | 荻野, 正雄 |
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| References | (14) Snyder, W., Tezaur, I., Christopher, W., Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method, arXiv, 2023, 2111.00224. (1) Raissi, M., Perdikaris, P., and Karniadakis, G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 2019, pp. 686-707. (9) 塩谷隆二, 矢川元基, 階層型領域分割法による1億自由度並列有限要素解析, 日本計算工学会論文集, 2001, 2001, Paper No. 20010024. (12) Kharazmi, E., Zhang, Z., Karmiadakis, G. E. M., hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Eng., 374-1, 2021, 113547. (18) Bjorstad, P.E. and Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Analysis, 23, 1986, pp. 1097-1120. (7) Smith, B. F., Bjorstad, P. E., and Gropp, W. D., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004. (17) Backstrom, B. and Backstrom, G., Simple Fields of Physics by Finite Element Analysis, GB Publishing, 2005. (3) 荻野正雄, 非一様度が低い点集合を用いたPINNによるポアソン方程式の予測モデル性能評価, 日本計算工学会論文集, 2024, 2024, Paper No. 20240003. (13) Shukla, K., Jagtap, A. D., Karniadakis, G. E., Parallel physics-informed neural networks via domain decomposition, J. Comput. Phys., 447-15, 2021, 110683. (2) Cuomo, S., Di Cola, V.S., Giampaolo, F., Rozza, G., Raissi, M., and Piccialli, F., Scientific machine learning through physics-informed neural networks: where we are and what's next, arXiv, 2022, 2201.05624v4. (5) Farhat, C. and Roux, F., Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2, 1994, pp. 1-124. (8) Tosseli, A. and Widlund, O., Domain Decomposition Methods-Algorithms and Theory, Springer, 2005. (10) 塩谷隆二, 金山寛, 田上大助, 荻野正雄, バランシング領域分割法による3次元大規模構造解析, 日本計算工学会論文集, 2000, 2000, Paper No. 20000017. (4) Yagawa, G. and Shioya, R., Parallel finite elements on a massively parallel computer with domain decomposition, Comput. Sys. Eng., 4, 1994, pp. 495-503. (6) Quarteroni, A. and Valli, A., Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999. (15) Hu, Z., Jagtap, A. D., Karniadakis, G. E., Augmented physics-informed neural networks (APINNs): A gating network-based soft domain decomposition methodology, Eng. Appl. Artif. Intell., 126-8, 2023, 107183. (11) Jagtap, A. D., Kharazmi, E., Karniadakis, G. E., Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Eng., 365-15, 2020, 113028. (16) Gu, L., Qin, S., Xu, L., Chen, R., Physics-informed neural networks with domain decomposition for the incompressible Navier-Stokes equations, Phys. Fluids, 36-2, 2024. (doi: 10.1063/5.0188830) |
| References_xml | – reference: (2) Cuomo, S., Di Cola, V.S., Giampaolo, F., Rozza, G., Raissi, M., and Piccialli, F., Scientific machine learning through physics-informed neural networks: where we are and what's next, arXiv, 2022, 2201.05624v4. – reference: (6) Quarteroni, A. and Valli, A., Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999. – reference: (17) Backstrom, B. and Backstrom, G., Simple Fields of Physics by Finite Element Analysis, GB Publishing, 2005. – reference: (10) 塩谷隆二, 金山寛, 田上大助, 荻野正雄, バランシング領域分割法による3次元大規模構造解析, 日本計算工学会論文集, 2000, 2000, Paper No. 20000017. – reference: (18) Bjorstad, P.E. and Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Analysis, 23, 1986, pp. 1097-1120. – reference: (8) Tosseli, A. and Widlund, O., Domain Decomposition Methods-Algorithms and Theory, Springer, 2005. – reference: (13) Shukla, K., Jagtap, A. D., Karniadakis, G. E., Parallel physics-informed neural networks via domain decomposition, J. Comput. Phys., 447-15, 2021, 110683. – reference: (9) 塩谷隆二, 矢川元基, 階層型領域分割法による1億自由度並列有限要素解析, 日本計算工学会論文集, 2001, 2001, Paper No. 20010024. – reference: (4) Yagawa, G. and Shioya, R., Parallel finite elements on a massively parallel computer with domain decomposition, Comput. Sys. Eng., 4, 1994, pp. 495-503. – reference: (7) Smith, B. F., Bjorstad, P. E., and Gropp, W. D., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004. – reference: (14) Snyder, W., Tezaur, I., Christopher, W., Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method, arXiv, 2023, 2111.00224. – reference: (1) Raissi, M., Perdikaris, P., and Karniadakis, G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 2019, pp. 686-707. – reference: (3) 荻野正雄, 非一様度が低い点集合を用いたPINNによるポアソン方程式の予測モデル性能評価, 日本計算工学会論文集, 2024, 2024, Paper No. 20240003. – reference: (5) Farhat, C. and Roux, F., Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2, 1994, pp. 1-124. – reference: (11) Jagtap, A. D., Kharazmi, E., Karniadakis, G. E., Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Eng., 365-15, 2020, 113028. – reference: (12) Kharazmi, E., Zhang, Z., Karmiadakis, G. E. M., hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Eng., 374-1, 2021, 113547. – reference: (16) Gu, L., Qin, S., Xu, L., Chen, R., Physics-informed neural networks with domain decomposition for the incompressible Navier-Stokes equations, Phys. Fluids, 36-2, 2024. (doi: 10.1063/5.0188830) – reference: (15) Hu, Z., Jagtap, A. D., Karniadakis, G. E., Augmented physics-informed neural networks (APINNs): A gating network-based soft domain decomposition methodology, Eng. Appl. Artif. Intell., 126-8, 2023, 107183. |
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| Snippet | PINNは偏微分方程式の初期境界値問題の誤差を損失関数に組み込んでニューラルネットワークを学習させる手法であり，多くの研究が報告されている．PINNによる予測解の精... |
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| Title | 2次元静磁場問題のNon-overlapping型領域分割法に基づくPINN |
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