allaPINNs: A physics-informed neural network with improvement of information representation and loss optimization for solving partial differential equations

ZHANG Zhaoyang; WANG Qingwang

doi:10.7498/aps.74.20250707

Abstract

Physics-informed neural networks (PINNs) have recently garnered significant attention as a meshless solution framework for solving partial differential equations (PDEs) in the context of AI-assisted scientific research (AI for Science). However, traditional PINNs exhibit certain limitations. On one hand, their network architecture, typically multilayer perceptrons (MLPs) with unidirectional information transfer, struggles to effectively capture key features embedded in sequential data, resulting in weak information characterization. On the other hand, the loss function of PINNs, a quadratic penalty function embedded with physical constraints, has an unconstrained and infinitely inflated penalty factor that affects the efficiency of the model’s training optimization search. To address these challenges, this paper proposes an improved PINN based on information representation and loss optimization, termed allaPINNs, which aims to enhance the model’s key feature extraction capability and training optimization search ability, thereby improving its accuracy and generalization for solving numerical solutions of PDEs. In terms of information characterization, allaPINNs introduces efficient linear attention (LA) to enhance the model’s ability to identify key features while reducing the computational complexity of dynamic weighting. In terms of loss optimization, allaPINNs reconstructs the objective loss function by introducing the augmented Lagrangian (AL) function, utilizing learnable Lagrangian multipliers and penalty factors to efficiently regulate the interaction of each loss residual term. The feasibility of allaPINNs is validated through four benchmark equations: Helmholtz, Black-Scholes, Burgers, and nonlinear Schrödinger. The results demonstrate that allaPINNs can effectively solve various PDEs of different complexities and exhibit excellent numerical solution prediction accuracy and generalization ability. Compared to the current state-of-the-art PINNs, the predictive accuracy is improved by one to two orders of magnitude.

Keywords:

Authors and contacts

Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming 650500, China

Corresponding author: WANG Qingwang, wangqingwang@kust.edu.cn

Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 62201237) and the Major Science and Technology Program of Yunnan Province, China (Grant Nos. 202202AD080013, 202302AG050009).

FullText HTML

References

[1]	Jin X, Cai S, Li H, Karniadakis G E 2021 J. Comput. Phys. 426 109951 Google Scholar
[2]	Roul P, Goura V P 2020 J. Comput. Appl. Math. 363 464 Google Scholar
[3]	Pu J C, Li J, Chen Y 2021 Chin. Phys. B 30 60202 Google Scholar
[4]	Cuomo S, Di Cola V S, Giampaolo F, Rozza G, Raissi M, Piccialli F 2022 J. Sci. Comput. 92 88 Google Scholar
[5]	Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T 2020 Comput. Methods Appl. Mech. Eng. 362 112790 Google Scholar
[6]	Taylor C A, Hughes T J, Zarins C K 1998 Comput. Methods Appl. Mech. Eng. 158 155 Google Scholar
[7]	Zhang Y 2009 Appl. Math. Comput. 215 524 Google Scholar
[8]	Van Hoecke L, Boeye D, Gonzalez-Quiroga A, Patience G S, Perreault P 2023 Can. J. Chem. Eng. 101 545 Google Scholar
[9]	Hasan F, Ali H, Arief H A 2025 Int. J. Appl. Comput. Math. 11 1 Google Scholar
[10]	Choo Y S, Choi N, Lee B C 2010 Appl. Math. Modell. 34 14 Google Scholar
[11]	Lawrence N D, Montgomery J 2024 R. Soc. Open Sci. 11 231130 Google Scholar
[12]	Si Z Z, Wang D L, Zhu B W, Ju Z T, Wang X P, Liu W, Malomed B A, Wang Y Y, Dai C Q 2024 Laser Photonics Rev. 18 2400097 Google Scholar
[13]	Fang Y, Han H B, Bo W B, Liu W, Wang B H, Wang Y Y, Dai C Q 2023 Opt. Lett. 48 779 Google Scholar
[14]	Li N, Xu S, Sun Y, Chen Q 2025 Nonlinear Dyn. 113 767 Google Scholar
[15]	Mouton L, Reiter F, Chen Y, Rebentrost P 2024 Phys. Rev. A 110 022612 Google Scholar
[16]	Zhu M, Feng S, Lin Y, Lu L 2023 Comput. Methods Appl. Mech. Eng. 416 116300 Google Scholar
[17]	Li X, Liu Z, Cui S, Luo C, Li C, Zhuang Z 2019 Comput. Methods Appl. Mech. Eng. 347 735 Google Scholar
[18]	Wang S, Teng Y, Perdikaris P 2021 SIAM J. Sci. Comput. 43 A3055 Google Scholar
[19]	Chew A W Z, He R, Zhang L 2025 Arch. Comput. Methods Eng. 32 399 Google Scholar
[20]	Bai J, Rabczuk T, Gupta A, Alzubaidi L, Gu Y 2023 Comput. Mech. 71 543 Google Scholar
[21]	Son S, Lee H, Jeong D, Oh K Y, Sun K H 2023 Adv. Eng. Inf. 57 102035 Google Scholar
[22]	方泽, 潘泳全, 戴栋, 张俊勃 2024 物理学报 73 145201 Google Scholar Fang Z, Pan Y Q, Dai D, Zhang J B 2024 Acta Phys. Sin. 73 145201 Google Scholar
[23]	Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686 Google Scholar
[24]	Hornik K 1991 Neural Networks 4 251 Google Scholar
[25]	Baydin A G, Pearlmutter B A, Radul A A, Siskind J M 2018 J. Mach. Learn. Res. 18 1
[26]	De Ryck T, Mishra S 2024 Acta Numer. 33 633 Google Scholar
[27]	Karniadakis G E, Kevrekidis I G, Lu L, Perdikaris P, Wang S, Yang L 2021 Nat. Rev. Phys. 3 422 Google Scholar
[28]	Ren P, Rao C, Liu Y, Wang J X, Sun H 2022 Comput. Methods Appl. Mech. Eng. 389 114399 Google Scholar
[29]	Lei L, He Y, Xing Z, Li Z, Zhou Y 2025 IEEE Trans. Ind. Inf. 21 5411 Google Scholar
[30]	Yuan B, Wang H, Heitor A, Chen X 2024 J. Comput. Phys. 515 113284 Google Scholar
[31]	Wang Y, Sun J, Bai J, Anitescu C, Eshaghi M S, Zhuang X, Rabczuk T, Liu Y 2025 Comput. Methods Appl. Mech. Eng. 433 117518 Google Scholar
[32]	Li L L, Zhang Y P, Wang G H, Xia K L 2025 Nat. Mach. Intell. 7 1346 Google Scholar
[33]	Jahani-Nasab M, Bijarchi M A 2024 Sci. Rep. 14 23836 Google Scholar
[34]	Yu J, Lu L, Meng X, Karniadakis G E 2022 Comput. Methods Appl. Mech. Eng. 393 114823 Google Scholar
[35]	Jiao Y, Lai Y, Lo Y, Wang Y, Yang Y 2024 Anal. Appl. 22 57 Google Scholar
[36]	Yang A, Xu S, Liu H, Li N, Sun Y 2025 Nonlinear Dyn. 113 1523 Google Scholar
[37]	Li Y, Zhou Z, Ying S 2022 J. Comput. Phys. 451 110884 Google Scholar
[38]	Jacot A, Gabriel F, Hongler C 2018 32nd Conference on Neural Information Processing Systems (NIPS 2018) Montreal, Canada, December 3–8, 2018 p8570
[39]	Xiang Z, Peng W, Liu X, Yao W 2022 Neurocomputing 496 11 Google Scholar
[40]	Tancik M, Srinivasan P, Mildenhall B, Fridovich-Keil S, Raghavan N, Singhal U, Ramamoorthi R, Barron J, Ng R 2020 34th Conference on Neural Information Processing Systems (NeurIPS 2020) Vancouver, Canada, December 6–12, 2020 p7537
[41]	Zhang Z, Wang Y, Tan S, Xia B, Luo Y 2025 Neurocomputing 625 129429 Google Scholar
[42]	Zhang W, Li H, Tang L, Gu X, Wang L, Wang L 2022 Acta Geotech. 17 1367 Google Scholar
[43]	Zhang Z, Wang Q, Zhang Y, Shen T, Zhang W 2025 Sci. Rep. 15 10523 Google Scholar
[44]	Cybenko G 1989 Math. Control Signals Syst. 2 303 Google Scholar
[45]	Wang C, Ma C, Zhou J 2014 J. Global Optim. 58 51 Google Scholar
[46]	Yi K, Zhang Q, Fan W, Wang S, Wang P, He H, An N, Lian D, Cao L, Niu Z 2023 Adv. Neural Inf. Process. Syst. 36 76656 Google Scholar
[47]	Durstewitz D, Koppe G, Thurm M I 2023 Nat. Rev. Neurosci. 24 693 Google Scholar
[48]	Chang G, Hu S, Huang H 2023 J. Supercomput. 79 6991 Google Scholar
[49]	Ocal H 2025 Arabian J. Sci. Eng. 50 1097 Google Scholar
[50]	Wu H C 2009 Eur. J. Oper. Res. 196 49 Google Scholar
[51]	Curtis F E, Jiang H, Robinson D P 2015 Math. Program. 152 201 Google Scholar
[52]	Sun D, Sun J, Zhang L 2008 Math. Program. 114 349 Google Scholar
[53]	Kanzow C, Steck D 2019 Math. Program. 177 425 Google Scholar
[54]	Rockafellar R T 2023 Math. Program. 198 159 Google Scholar
[55]	Dampfhoffer M, Mesquida T, Valentian A, Anghel L 2023 IEEE Trans. Neural Networks Learn. Syst. 35 11906 Google Scholar
[56]	Humbird K D, Peterson J L, McClarren R G 2018 IEEE Trans. Neural Networks Learn. Syst. 30 1286 Google Scholar
[57]	Zhang Z, Wang Q, Zhang Y, Shen T 2025 Digital Signal Process. 156 104766 Google Scholar
[58]	Zhou P, Xie X, Lin Z, Yan S 2024 IEEE Trans. Pattern Anal. Mach. Intell. 46 6486 Google Scholar
[59]	Rather I H, Kumar S, Gandomi A H 2024 Artif. Intell. Rev. 57 226 Google Scholar
[60]	Thulasidharan K, Priya N V, Monisha S, Senthilvelan M 2024 Phys. Lett. A 511 129551 Google Scholar
[61]	Son H, Cho S W, Hwang H J 2023 Neurocomputing 548 126424 Google Scholar
[62]	Song Y, Wang H, Yang H, Taccari M L, Chen X 2024 J. Comput. Phys. 501 112781 Google Scholar

Cited By

图 1 allaPINNs网络结构

Figure 1. allaPINNs network structure.

DownLoad: Full-Size Img PowerPoint

图 2 使用allaPINNs求解基准方程的预测解和解析解对比结果

Figure 2. Comparison of predicted and analytical solutions for solving the benchmark equations using allaPINNs.

DownLoad: Full-Size Img PowerPoint

图 3 使用allaPINNs在$ t = 0,\; 0.3,\; 0.66,\; 0.88 $四个时刻求解非线性Schrödinger方程的振幅快照

Figure 3. Snapshots of amplitudes for solving nonlinear Schrödinger equation at four moments $ t = 0,\; 0.3,\; 0.66,\; 0.88 $ using allaPINNs

DownLoad: Full-Size Img PowerPoint

图 4 使用allaPINNs求解基准方程的逐点绝对误差分布

Figure 4. Pointwise absolute error distributions for solving benchmark equations using allaPINNs.

DownLoad: Full-Size Img PowerPoint

图 5 最优拉格朗日乘子$ \lambda_{*} $的频率直方图, 其中Helmholtz方程包含$ \lambda_{1}^{*} $, Black-Scholes方程包含$ \lambda_{1}^{*}{\text{—}}\lambda_{3}^{*} $, Burgers方程包含$ \lambda_{1}^{*} $和$ \lambda_{2}^{*} $, 非线性Schrödinger方程包含$ \lambda_{1}^{*}{\text{—}}\lambda_{6}^{*} $

Figure 5. Frequency histogram of the optimal Lagrangian multipliers $ \lambda^{*} $, where the Helmholtz equation contains $ \lambda_{1}^{*} $, the Black-Scholes equation contains $ \lambda_{1}^{*}-\lambda_{3}^{*} $, the Burgers equation contains $ \lambda_{1}^{*}, \;\lambda_{2}^{*} $, and nonlinear Schrödinger equation contains $ \lambda_{1}^{*}-\lambda_{6}^{*} $.

DownLoad: Full-Size Img PowerPoint

图 6 惩罚因子σ的学习过程, 其中(a), (b), (c), (d)中的$ \sigma^{*} $分别位于第48665, 49420, 43467和48933轮迭代, 对应的数值分别为2517.21, 35.45, 5.02和52.08

Figure 6. Learning process of the penalty factor σ, where $ \sigma^{*} $ for (a), (b), (c), (d) are located at rounds 48665, 49420, 43467, and 48933, corresponding to values of 2517.21, 35.45, 5.02, and 52.08, respectively.

DownLoad: Full-Size Img PowerPoint

图 7 allaPINNs在训练过程中的$ L^{2} $相对泛化误差

Figure 7. The $ L^{2} $ relative generalization error of allaPINNs during training.

DownLoad: Full-Size Img PowerPoint

表 1 allaPINNs模型和当前先进PINNs模型求解四个基准方程实验的$ L^{2} $相对误差对比结果

Table 1. Comparison of $ L^{2} $ relative errors between allaPINNs model and current state-of-the-art PINNs model for solving the four benchmark equations.

	物理信息神经求解器
求解方程	PINNs^[23]	AL-PINNs^[61]	f-PICNN^[30]	KINN^[31]	allaPINNs (本文)
Helmholtz	5.63 × 10^–2	1.82 × 10^–3	2.51 × 10^–3	1.08 × 10^–3	8.06 × 10^–4
Black-Scholes	7.18 × 10^–2	7.41 × 10^–3	5.24 × 10^–3	4.35 × 10^–3	3.48 × 10^–4
Burgers	7.04 × 10^–2	3.39 × 10^–3	2.49 × 10^–3	3.02 × 10^–3	8.31 × 10^–4
非线性Schrödinger	2.09 × 10^–2	1.55 × 10^–3	4.18 × 10^–3	1.37 × 10^–3	6.71 × 10^–4

DownLoad: CSV

表 2 allaPINNs模型在不同网络结构和损失函数条件下的平均$ L^{2} $相对误差对比结果

Table 2. Comparison of mean $ L^{2} $ relative errors of allaPINNs model with different network structure and loss function conditions.

网络结构	基准方程平均$ L^{2} $相对误差	损失函数	基准方程平均$ L^{2} $相对误差
MLPs	7.26 × 10^–2	罚函数	8.55 × 10^–3
CNN	3.09 × 10^–3	罚函数 (高斯噪声)	1.63 × 10^–3
LA	6.61 × 10^–4	AL	6.61 × 10^–4

DownLoad: CSV

表 B1 各对比模型中超参数配置和求解基准方程时间消耗

Table B1. Hyperparameter configuration and time consumption for solving the benchmark equations in each comparison model.

	物理信息神经求解器
对比指标	PINNs^[23]	AL-PINNs^[61]	f-PICNN^[30]	KINN^[31]	allaPINNs (本文)
网络层数	8	8	6	5	5
神经元数量	200	256	128	80	64
参数学习率	1 × 10^–3	1 × 10^–4	1 × 10^–4	1 × 10^–4	1 × 10^–3
求解Helmholtz时间消耗/s	1476	1520	1529	1606	1513
求解Black-Scholes时间消耗/s	1538	1585	1604	1714	1624
求解Burgers时间消耗/s	1519	1574	1593	1636	1588
求解非线性Schrödinger时间消耗/s	1545	1628	1650	1745	1661

DownLoad: CSV

[1]	Jin X, Cai S, Li H, Karniadakis G E 2021 J. Comput. Phys. 426 109951 Google Scholar
[2]	Roul P, Goura V P 2020 J. Comput. Appl. Math. 363 464 Google Scholar
[3]	Pu J C, Li J, Chen Y 2021 Chin. Phys. B 30 60202 Google Scholar
[4]	Cuomo S, Di Cola V S, Giampaolo F, Rozza G, Raissi M, Piccialli F 2022 J. Sci. Comput. 92 88 Google Scholar
[5]	Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T 2020 Comput. Methods Appl. Mech. Eng. 362 112790 Google Scholar
[6]	Taylor C A, Hughes T J, Zarins C K 1998 Comput. Methods Appl. Mech. Eng. 158 155 Google Scholar
[7]	Zhang Y 2009 Appl. Math. Comput. 215 524 Google Scholar
[8]	Van Hoecke L, Boeye D, Gonzalez-Quiroga A, Patience G S, Perreault P 2023 Can. J. Chem. Eng. 101 545 Google Scholar
[9]	Hasan F, Ali H, Arief H A 2025 Int. J. Appl. Comput. Math. 11 1 Google Scholar
[10]	Choo Y S, Choi N, Lee B C 2010 Appl. Math. Modell. 34 14 Google Scholar
[11]	Lawrence N D, Montgomery J 2024 R. Soc. Open Sci. 11 231130 Google Scholar
[12]	Si Z Z, Wang D L, Zhu B W, Ju Z T, Wang X P, Liu W, Malomed B A, Wang Y Y, Dai C Q 2024 Laser Photonics Rev. 18 2400097 Google Scholar
[13]	Fang Y, Han H B, Bo W B, Liu W, Wang B H, Wang Y Y, Dai C Q 2023 Opt. Lett. 48 779 Google Scholar
[14]	Li N, Xu S, Sun Y, Chen Q 2025 Nonlinear Dyn. 113 767 Google Scholar
[15]	Mouton L, Reiter F, Chen Y, Rebentrost P 2024 Phys. Rev. A 110 022612 Google Scholar
[16]	Zhu M, Feng S, Lin Y, Lu L 2023 Comput. Methods Appl. Mech. Eng. 416 116300 Google Scholar
[17]	Li X, Liu Z, Cui S, Luo C, Li C, Zhuang Z 2019 Comput. Methods Appl. Mech. Eng. 347 735 Google Scholar
[18]	Wang S, Teng Y, Perdikaris P 2021 SIAM J. Sci. Comput. 43 A3055 Google Scholar
[19]	Chew A W Z, He R, Zhang L 2025 Arch. Comput. Methods Eng. 32 399 Google Scholar
[20]	Bai J, Rabczuk T, Gupta A, Alzubaidi L, Gu Y 2023 Comput. Mech. 71 543 Google Scholar
[21]	Son S, Lee H, Jeong D, Oh K Y, Sun K H 2023 Adv. Eng. Inf. 57 102035 Google Scholar
[22]	方泽, 潘泳全, 戴栋, 张俊勃 2024 物理学报 73 145201 Google Scholar Fang Z, Pan Y Q, Dai D, Zhang J B 2024 Acta Phys. Sin. 73 145201 Google Scholar
[23]	Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686 Google Scholar
[24]	Hornik K 1991 Neural Networks 4 251 Google Scholar
[25]	Baydin A G, Pearlmutter B A, Radul A A, Siskind J M 2018 J. Mach. Learn. Res. 18 1
[26]	De Ryck T, Mishra S 2024 Acta Numer. 33 633 Google Scholar
[27]	Karniadakis G E, Kevrekidis I G, Lu L, Perdikaris P, Wang S, Yang L 2021 Nat. Rev. Phys. 3 422 Google Scholar
[28]	Ren P, Rao C, Liu Y, Wang J X, Sun H 2022 Comput. Methods Appl. Mech. Eng. 389 114399 Google Scholar
[29]	Lei L, He Y, Xing Z, Li Z, Zhou Y 2025 IEEE Trans. Ind. Inf. 21 5411 Google Scholar
[30]	Yuan B, Wang H, Heitor A, Chen X 2024 J. Comput. Phys. 515 113284 Google Scholar
[31]	Wang Y, Sun J, Bai J, Anitescu C, Eshaghi M S, Zhuang X, Rabczuk T, Liu Y 2025 Comput. Methods Appl. Mech. Eng. 433 117518 Google Scholar
[32]	Li L L, Zhang Y P, Wang G H, Xia K L 2025 Nat. Mach. Intell. 7 1346 Google Scholar
[33]	Jahani-Nasab M, Bijarchi M A 2024 Sci. Rep. 14 23836 Google Scholar
[34]	Yu J, Lu L, Meng X, Karniadakis G E 2022 Comput. Methods Appl. Mech. Eng. 393 114823 Google Scholar
[35]	Jiao Y, Lai Y, Lo Y, Wang Y, Yang Y 2024 Anal. Appl. 22 57 Google Scholar
[36]	Yang A, Xu S, Liu H, Li N, Sun Y 2025 Nonlinear Dyn. 113 1523 Google Scholar
[37]	Li Y, Zhou Z, Ying S 2022 J. Comput. Phys. 451 110884 Google Scholar
[38]	Jacot A, Gabriel F, Hongler C 2018 32nd Conference on Neural Information Processing Systems (NIPS 2018) Montreal, Canada, December 3–8, 2018 p8570
[39]	Xiang Z, Peng W, Liu X, Yao W 2022 Neurocomputing 496 11 Google Scholar
[40]	Tancik M, Srinivasan P, Mildenhall B, Fridovich-Keil S, Raghavan N, Singhal U, Ramamoorthi R, Barron J, Ng R 2020 34th Conference on Neural Information Processing Systems (NeurIPS 2020) Vancouver, Canada, December 6–12, 2020 p7537
[41]	Zhang Z, Wang Y, Tan S, Xia B, Luo Y 2025 Neurocomputing 625 129429 Google Scholar
[42]	Zhang W, Li H, Tang L, Gu X, Wang L, Wang L 2022 Acta Geotech. 17 1367 Google Scholar
[43]	Zhang Z, Wang Q, Zhang Y, Shen T, Zhang W 2025 Sci. Rep. 15 10523 Google Scholar
[44]	Cybenko G 1989 Math. Control Signals Syst. 2 303 Google Scholar
[45]	Wang C, Ma C, Zhou J 2014 J. Global Optim. 58 51 Google Scholar
[46]	Yi K, Zhang Q, Fan W, Wang S, Wang P, He H, An N, Lian D, Cao L, Niu Z 2023 Adv. Neural Inf. Process. Syst. 36 76656 Google Scholar
[47]	Durstewitz D, Koppe G, Thurm M I 2023 Nat. Rev. Neurosci. 24 693 Google Scholar
[48]	Chang G, Hu S, Huang H 2023 J. Supercomput. 79 6991 Google Scholar
[49]	Ocal H 2025 Arabian J. Sci. Eng. 50 1097 Google Scholar
[50]	Wu H C 2009 Eur. J. Oper. Res. 196 49 Google Scholar
[51]	Curtis F E, Jiang H, Robinson D P 2015 Math. Program. 152 201 Google Scholar
[52]	Sun D, Sun J, Zhang L 2008 Math. Program. 114 349 Google Scholar
[53]	Kanzow C, Steck D 2019 Math. Program. 177 425 Google Scholar
[54]	Rockafellar R T 2023 Math. Program. 198 159 Google Scholar
[55]	Dampfhoffer M, Mesquida T, Valentian A, Anghel L 2023 IEEE Trans. Neural Networks Learn. Syst. 35 11906 Google Scholar
[56]	Humbird K D, Peterson J L, McClarren R G 2018 IEEE Trans. Neural Networks Learn. Syst. 30 1286 Google Scholar
[57]	Zhang Z, Wang Q, Zhang Y, Shen T 2025 Digital Signal Process. 156 104766 Google Scholar
[58]	Zhou P, Xie X, Lin Z, Yan S 2024 IEEE Trans. Pattern Anal. Mach. Intell. 46 6486 Google Scholar
[59]	Rather I H, Kumar S, Gandomi A H 2024 Artif. Intell. Rev. 57 226 Google Scholar
[60]	Thulasidharan K, Priya N V, Monisha S, Senthilvelan M 2024 Phys. Lett. A 511 129551 Google Scholar
[61]	Son H, Cho S W, Hwang H J 2023 Neurocomputing 548 126424 Google Scholar
[62]	Song Y, Wang H, Yang H, Taccari M L, Chen X 2024 J. Comput. Phys. 501 112781 Google Scholar

[1]	SUN Kangsheng, HAN Chao, QIN Haifeng, GU Tao, LI Wei, YU Cheng. Fast generation algorithm of high-quality holograms based on attention convolutional neural network. Acta Physica Sinica, 2025, 74(8): 084203. doi: 10.7498/aps.74.20241713
[2]	WANG Xiang, ZHOU Yishen, ZHANG Xuange, CHEN Xihao. Convolutional network single-pixel imaging with fusion attention mechanism. Acta Physica Sinica, 2025, 74(8): 084202. doi: 10.7498/aps.74.20250010
[3]	LI Jun, SU Jin, HAN Xiaoxiang, ZHU Weijie, YANG Ruixia, ZHANG Haiyang, YAN Xiang’an, ZHANG Yunjie, WANG Feiran. Research on pulse transmission dynamics in optical fiber based on intelligent algorithms. Acta Physica Sinica, 2025, 74(6): 060201. doi: 10.7498/aps.74.20241473
[4]	LIU Ming, ZHANG Siqi, LI Hong. Dynamic analysis of shortcut to adiabaticity based on physical information neural network. Acta Physica Sinica, 2025, 74(11): 114204. doi: 10.7498/aps.74.20250147
[5]	WANG Yuduo, CHEN Jiaxin, LI Biao. Solving nonlinear Schrödinger equations and parameter discovery via extended mixed-training physics-informed neural networks. Acta Physica Sinica, 2025, 74(16): 160201. doi: 10.7498/aps.74.20250422
[6]	Fang Ze, Pan Yong-Quan, Dai Dong, Zhang Jun-Bo. Physics-informed neural networks based on source term decoupled and its application in discharge plasma simulation. Acta Physica Sinica, 2024, 73(14): 145201. doi: 10.7498/aps.73.20240343
[7]	Tian Shi-Fang, Li Biao. Solving complex nonlinear problems based on gradient-optimized physics-informed neural networks. Acta Physica Sinica, 2023, 72(10): 100202. doi: 10.7498/aps.72.20222381
[8]	Fang Bo-Lang, Wang Jian-Guo, Feng Guo-Bin. Calculation of spot entroid based on physical informed neural networks. Acta Physica Sinica, 2022, 71(20): 200601. doi: 10.7498/aps.71.20220670
[9]	Huang Wei-Jian, Li Yong-Tao, Huang Yuan. Prediction of chaotic time series using hybrid neural network and attention mechanism. Acta Physica Sinica, 2021, 70(1): 010501. doi: 10.7498/aps.70.20200899
[10]	Huang Liang, Li Jian-Yuan. Modeling and failure monitor of Li-ion battery based on single particle model and partial difference equations. Acta Physica Sinica, 2015, 64(10): 108202. doi: 10.7498/aps.64.108202
[11]	Li Huan, Wang You-Guo. Noise-enhanced information transmission of a non-linear multilevel threshold neural networks system. Acta Physica Sinica, 2014, 63(12): 120506. doi: 10.7498/aps.63.120506
[12]	Sudao Bilige, Wang Xiao-Min, Wuyun Morigen. Application of the symmetry classification to the boundary value problem of nonlinear partial differential equations. Acta Physica Sinica, 2014, 63(4): 040201. doi: 10.7498/aps.63.040201
[13]	He Yu-Bo, Lin Xiao-Yan, Dong Xiao-Liang. Use of lattice Boltzmann method to simulate 2-D partial differential equation. Acta Physica Sinica, 2013, 62(19): 194701. doi: 10.7498/aps.62.194701
[14]	Hou Xiang-Lin, Zhai Zhong-Hai, Zheng Li, Liu Tie-Lin. Layered optimization algorithm about a kind of nonlinear partial differential equation with initial-boundary value problem. Acta Physica Sinica, 2012, 61(1): 010201. doi: 10.7498/aps.61.010201
[15]	Hou Xiang-Lin, Zheng Xi-Jian, Zhang Liang, Liu Tie-Lin. Study on derivation and optimization algorithm about thin plate bending large deformation higher-order nonlinear partial differential equations. Acta Physica Sinica, 2012, 61(18): 180201. doi: 10.7498/aps.61.180201
[16]	Liu Tie-Lin, Zhai Zhong-Hai, Hou Xiang-Lin. Study and application of optimization algorithm about nonlinear partial differential equations with boundary value problem. Acta Physica Sinica, 2011, 60(9): 090202. doi: 10.7498/aps.60.090202
[17]	Xie Yuan-Xi, Tang Jia-Shi. A note on paper “a simple fast method in finding the analytical solutions to a class of nonlinear partial differential equations”. Acta Physica Sinica, 2005, 54(3): 1036-1038. doi: 10.7498/aps.54.1036
[18]	Xie Yuan-Xi, Tang Jia-Shi. A simple fast method in finding the analytical solutions to a class of nonlinear partial differential equations. Acta Physica Sinica, 2004, 53(9): 2828-2830. doi: 10.7498/aps.53.2828
[19]	Lu Jing, Yan Jia-Ren. . Acta Physica Sinica, 2002, 51(7): 1428-1433. doi: 10.7498/aps.51.1428
[20]	. . Acta Physica Sinica, 1963, 19(8): 538-540. doi: 10.7498/aps.19.538

Catalog

Vol. 74, Issue 18 - 2025-09-20

Metrics

Abstract views: 647
PDF Downloads: 23
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板