Pith. sign in

REVIEW 31 references

LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2506.17582 v1 pith:5KJU4ZYW submitted 2025-06-21 cs.LG physics.comp-ph

Jing Wang , Biao Chen , Hairun Xie , Rui Wang , Yifan Xia , Jifa Zhang , Hui Xu This is my paper · ORCID
classification cs.LGphysics.comp-ph
keywords lfr-pinoneuralparameterphysics-informedreductionsolutionwhilecompared
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
read the original abstract

Physics-informed neural operators have emerged as a powerful paradigm for solving parametric partial differential equations (PDEs), particularly in the aerospace field, enabling the learning of solution operators that generalize across parameter spaces. However, existing methods either suffer from limited expressiveness due to fixed basis/coefficient designs, or face computational challenges due to the high dimensionality of the parameter-to-weight mapping space. We present LFR-PINO, a novel physics-informed neural operator that introduces two key innovations: (1) a layered hypernetwork architecture that enables specialized parameter generation for each network layer, and (2) a frequency-domain reduction strategy that significantly reduces parameter count while preserving essential spectral features. This design enables efficient learning of a universal PDE solver through pre-training, capable of directly handling new equations while allowing optional fine-tuning for enhanced precision. The effectiveness of this approach is demonstrated through comprehensive experiments on four representative PDE problems, where LFR-PINO achieves 22.8%-68.7% error reduction compared to state-of-the-art baselines. Notably, frequency-domain reduction strategy reduces memory usage by 28.6%-69.3% compared to Hyper-PINNs while maintaining solution accuracy, striking an optimal balance between computational efficiency and solution fidelity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 1 canonical work pages

  1. [1]

    J. Li, X. Du, and J.R. Martins,Machine learning in aerodynamic shape optimiza- tion, Progress in Aerospace Sciences 134 (2022), p. 100849

    2022

  2. [2]

    J. Han, A. Jentzen, and W. E,Solving high-dimensional partial differential equa- tions using deep learning, Proceedings of the National Academy of Sciences 115 (2018), pp. 8505–8510

    2018

  3. [3]

    Zienkiewicz, R

    O. Zienkiewicz, R. Taylor, and J. Zhu,Chapter 1 - the standard discrete sys- tem and origins of the finite element method, inThe Finite Element Method: its Basis and Fundamentals (Seventh Edition), O. Zienkiewicz, R. Taylor, and J. Zhu, eds., seventh edition ed., Butterworth-Heinemann, Oxford, 2013, pp. 1–20, Available athttps://www.sciencedirect.com/scie...

    2013

  4. [4]

    Godunov and I

    S.K. Godunov and I. Bohachevsky,Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Matematiˇ ceskij sbornik 47 (1959), pp. 271–306

    1959

  5. [5]

    Eymard, T

    R. Eymard, T. Gallou¨ et, and R. Herbin,Finite volume methods, Handbook of numerical analysis 7 (2000), pp. 713–1018. 19

    2000

  6. [6]

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G.E. Karniadakis,Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature machine intelligence 3 (2021), pp. 218–229

    2021

  7. [7]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar,Fourier neural operator for parametric partial differential equations (2021)

    2021

  8. [8]

    S. Wang, H. Wang, and P. Perdikaris,Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets. Available athttp: //arxiv.org/abs/2103.10974

    arXiv

Show all 31 references
  1. [9]

    Huang, Z

    X. Huang, Z. Ye, H. Liu, B. Shi, Z. Wang, K. Yang, Y. Li, B. Weng, M. Wang, H. Chu, F. Yu, B. Hua, L. Chen, and B. Dong,Meta-Auto-Decoder for Solving Para- metric Partial Differential Equations, arXiv e-prints (2021), arXiv:2111.08823

    2021 arXiv

  2. [10]

    Belbute-Peres, F

    F.d.A. Belbute-Peres, F. Sha, and Y.f. Chen,HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks

  3. [11]

    J. Xu, Y. Zhang, and Y. Xiao,Training behavior of deep neural network in fre- quency domain(2019)

    2019

  4. [12]

    Xu,Understanding training and generalization in deep learning by fourier anal- ysis, CoRR abs/1808.04295 (2018), Available athttp://arxiv.org/abs/1808

    J. Xu,Understanding training and generalization in deep learning by fourier anal- ysis, CoRR abs/1808.04295 (2018), Available athttp://arxiv.org/abs/1808. 04295

    2018 arXiv

  5. [13]

    LeCun, Y

    Y. LeCun, Y. Bengio, and G. Hinton,Deep learning, nature 521 (2015), pp. 436– 444

    2015

  6. [14]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. Karniadakis,Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019), pp. 686–707, Available athttps://w...

    2019

  7. [15]

    Sirignano and K

    J. Sirignano and K. Spiliopoulos,Dgm: A deep learning algorithm for solving partial differential equations, Journal of computational physics 375 (2018), pp. 1339–1364

    2018

  8. [16]

    Yu,et al.,The deep ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (2018), pp

    B. Yu,et al.,The deep ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (2018), pp. 1–12

    2018

  9. [17]

    C. Rao, P. Ren, Q. Wang, O. Buyukozturk, H. Sun, and Y. Liu,Encoding physics to learn reaction–diffusion processes, Nature Machine Intelligence 5 (2023), pp. 765–779

    2023

  10. [18]

    Raissi, A

    M. Raissi, A. Yazdani, and G.E. Karniadakis,Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367 (2020), pp. 1026– 1030

    2020

  11. [19]

    H. Xu, W. Zhang, and Y. Wang,Explore missing flow dynamics by physics- informed deep learning: The parameterized governing systems, Physics of Fluids 33 (2021)

    2021

  12. [20]

    Z. Long, Y. Lu, X. Ma, and B. Dong,Pde-net: Learning pdes from data, inInter- national conference on machine learning, PMLR. 2018, pp. 3208–3216

    2018

  13. [21]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar,Neural operator: Graph kernel network for partial differential equations, arXiv preprint arXiv:2003.03485 (2020)

    2020 arXiv

  14. [22]

    Z. Deng, J. Wang, H. Liu, H. Xie, B. Li, M. Zhang, T. Jia, Y. Zhang, Z. Wang, and B. Dong,Prediction of transonic flow over supercritical airfoils using geometric- encoding and deep-learning strategies, Physics of Fluids 35 (2023)

    2023

  15. [23]

    Penwarden, S

    M. Penwarden, S. Zhe, A. Narayan, and R.M. Kirby,A metalearning approach 20 for physics-informed neural networks (PINNs): Application to parameterized PDEs477, p. 111912, Available athttps://www.sciencedirect.com/science/ article/pii/S0021999123000074

  16. [24]

    X. Liu, X. Zhang, W. Peng, W. Zhou, and W. Yao,A novel meta-learning initial- ization method for physics-informed neural networks34, pp. 14511–14534, Avail- able athttps://link.springer.com/10.1007/s00521-022-07294-2

    doi

  17. [25]

    Y. Guo, X. Cao, J. Song, and H. Leng,MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization33, p. 020203, Available athttps://iopscience. iop.org/article/10.1088/1674-1056/ad0bf4

    doi

  18. [26]

    Zou and G.E

    Z. Zou and G.E. Karniadakis,L-hydra: Multi-head physics-informed neural net- works, arXiv preprint arXiv:2301.02152 (2023)

    2023

  19. [27]

    D. Ha, A. Dai, and Q.V. Le,Hypernetworks, arXiv preprint arXiv:1609.09106 (2016)

    2016 arXiv

  20. [28]

    K. Shi, X. Zhou, and S. Gu,Improved Implicit Neural Representation with Fourier Reparameterized Training, inProceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition. 2024, pp. 25985–25994

    2024

  21. [29]

    Xiang, W

    Z. Xiang, W. Peng, X. Liu, and W. Yao,Self-adaptive loss balanced physics- informed neural networks, Neurocomputing 496 (2022), pp. 11–34

    2022

  22. [30]

    J. Yu, L. Lu, X. Meng, and G.E. Karniadakis,Gradient-enhanced physics- informed neural networks for forward and inverse pde problems, Computer Meth- ods in Applied Mechanics and Engineering 393 (2022), p. 114823

    2022

  23. [31]

    Iserles,A first course in the numerical analysis of differential equations, 44, Cambridge university press, 2009

    A. Iserles,A first course in the numerical analysis of differential equations, 44, Cambridge university press, 2009. Appendix A. Detailed proof of Theorem 2 Theorem 2(Low-frequency Gradient Theorem).For thel-th layer weight matrix W(l) =Λ (l)B(l), whereΛ (l) ∈R d×M andB (l) ∈R...

    2009

Pith tools