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Machine Learning / Deep Learning for Partial Differential Equations (PDEs) Solvers

Recently, there are a growing number of papers trying to solve PDEs with Machine Learning. This respository is trying to collect and sort papers, blogs, videos, and any format materials in this field.

Model Zoo

Model Relevant Papers Link Notes
HiDeNN Saha, Sourav, et al. "Hierarchical Deep Learning Neural Network (HiDeNN): An artificial intelligence (AI) framework for computational science and engineering." Computer Methods in Applied Mechanics and Engineering 373 (2021): 113452. Paper
HiTSs Liu, Yuying, J. Nathan Kutz, and Steven L. Brunton. "Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers." arXiv preprint arXiv:2008.09768 (2020). Paper, Code, Video
Kochkov, Dmitrii, et al. "Machine learning accelerated computational fluid dynamics." arXiv preprint arXiv:2102.01010 (2021). Paper Google
Fourier Neural Operator Li, Zongyi, et al. "Fourier neural operator for parametric partial differential equations." arXiv preprint arXiv:2010.08895 (2020). Paper, Code, Video
PINNs Raissi, Maziar, Paris Perdikaris, and George E. 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): 686-707; Paper, Code, Video
Raissi, Maziar, Alireza Yazdani, and George Em Karniadakis. "Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations." Science 367.6481 (2020): 1026-1030. Paper Science
Ling, Julia, Andrew Kurzawski, and Jeremy Templeton. "Reynolds averaged turbulence modelling using deep neural networks with embedded invariance." Journal of Fluid Mechanics 807 (2016): 155-166. Paper, Code Pure data
K. Duraisamy, G. Iaccarino, and H. Xiao, Turbulence modeling in the age of data, Annual Review of Fluid Mechanics 51, 357 (2019). Paper Pure data, Review
Maulik, Romit, et al. "Subgrid modelling for two-dimensional turbulence using neural networks." Journal of Fluid Mechanics 858 (2019): 122-144. Paper
Beck, Andrea, David Flad, and Claus-Dieter Munz. "Deep neural networks for data-driven LES closure models." Journal of Computational Physics 398 (2019): 108910. Paper
Lusch, Bethany, J. Nathan Kutz, and Steven L. Brunton. "Deep learning for universal linear embeddings of nonlinear dynamics." Nature communications 9.1 (2018): 1-10. Paper Nature communications
Freund, Jonathan B., Jonathan F. MacArt, and Justin Sirignano. "DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)." arXiv preprint arXiv:1911.09145 (2019). Paper
Um, Kiwon, et al. "Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers." arXiv preprint arXiv:2007.00016 (2020). Paper
DeepONet Lu, Lu, Pengzhan Jin, and George Em Karniadakis. "Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators." arXiv preprint arXiv:1910.03193 (2019). Paper, Video
PINN S. Alkhadhr, X. Liu, & M. Almekkawy. "Modeling of the forward wave propagation using physics-informed neural networks." IEEE International Ultrasonics Symposium (IUS), pp. 1–4 (2021). Paper
PINN C. Martin, A. Oved, R. Chowdhury, E. Ullmann, N. Peters, A. Bharath, & M. Varela. "EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks." Frontiers in cardiovascular medicine, 2179. (2022) Paper
PINN Y. Xue, Y. Li, K. Zhang, & F. Yang. "A physics-inspired neural network to solve partial differential equations - application in diffusion-induced stress." Physical Chemistry Chemical Physics, 24(13), 7937-7949 (2022) Paper
PINN A. Sacchetti, B. Bachmann, K. Löffel, U. Künzi, & B. Paoli. "Neural networks to solve partial differential equations: A comparison with finite elements." IEEE Access, 10, 32271-32279. (2022) Paper
PINN J. Yu, L. Lu, X. Meng, & G. Karniadakis. "Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems." Computer Methods in Applied Mechanics and Engineering, 393, 114823. (2022) Paper
PINN Y. Lu, G. Mei, & F. Piccialli. "A deep learning approach for predicting two-dimensional soil consolidation using physics-informed neural networks." arXiv preprint arXiv:2205.05710. (2022) Paper
PINN L. Guo, H. Wu, X. Yu, & T. Zhou. "Monte Carlo PINNs: Deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations." arXiv preprint arXiv:2203.08501. (2022) Paper
PINN Y. Wang, X. Han, C. Chang, D. Zha, U. Braga-Neto, & X. Hu. "Auto-PINN: Understanding and optimizing physics-informed neural architecture." arXiv preprint arXiv:2205.13748. (2022) Paper
PINN F. Torres, M. Negri, M. Nagy-Huber, M. Samarin, & V. Roth. "Mesh-free Eulerian physics-informed neural networks." arXiv preprint arXiv:2206.01545. (2022) Paper
PINN N. Dhamirah Mohamad, A. Yousif, N. Shaari, H. Mustafa, S. Abdul Karim, A. Shafie, & M. Izzatullah. "Heat transfer modeling with physics-informed neural network." Intelligent Systems Modeling and Simulation II: Machine Learning, Neural Networks, Efficient Numerical Algorithm and Statistical Methods, pp. 25-35, Cham: Springer International Publishing. (2022) Paper
PINN A. Serebrennikova, R. Teubler, L. Hoffellner, E. Leitner, U. Hirn, & K. Zojer. "Transport of organic volatiles through paper: Physics-informed neural networks for solving inverse and forward problems." Transport in Porous Media, 1-24. (2022) Paper
PINN C. McDevitt, E. Fowler, & S. Roy. "Physics-constrained deep learning of incompressible cavity flows." arXiv preprint arXiv:2211.06375. (2022) Paper
PINN S. Carney, A. Gangal, & L. Kim. "Physics informed neural networks for elliptic equations with oscillatory differential operators." arXiv preprint arXiv:2212.13531. (2022) Paper
PINN L. Sliwinski, & G. Rigas. "Mean flow reconstruction of unsteady flows using physics-informed neural networks."Data-Centric Engineering, 4, p.e4. (2023) Paper
PINN F. Pioch, J. Harmening, A. Müller, F. Peitzmann, D. Schramm, & O. Moctar. "Turbulence modeling for physics-informed neural networks: Comparison of different RANS models for the backward-facing step flow." Fluids, 8(2), p.43. (2023) Paper
PINN P. Sharma, L. Evans, M. Tindall, & P. Nithiarasu. "Stiff-PDEs and physics-informed neural networks." Archives of Computational Methods in Engineering. (2023) Paper
PINN S. Alkhadhr, & M. Almekkawy. "Wave equation modeling via physics-informed neural networks: models of soft and hard constraints for initial and boundary conditions." Sensors, 23(5), 2792. (2023) Paper
PINN J. Yao, C. Su, Z. Hao, S. Liu, H. Su, and J. Zhu. "Multiadam: Parameter-wise scale-invariant optimizer for multiscale training of physics-informed neural networks." arXiv preprint arxiv:2306.02816 Paper
DeepONet C. Lin, Z. Li, L. Lu, S. Cai, M. Maxey, & G. Karniadakis. "Operator learning for predicting multiscale bubble growth dynamics." The Journal of Chemical Physics, 154(10), 104118. (2021) Paper
DeepM&MNet S. Cai, Z. Wang, L. Lu, T. Zaki, & G. Karniadakis. "DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks." Journal of Computational Physics, 436, 110296. (2021) Paper
DeepM&MNet Z. Mao, L. Lu, O. Marxen, T. Zaki, & G. Karniadakis. "DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators." Journal of Computational Physics, 447, 110698. (2021) Paper
MIONet P. Jin, S. Meng, & L. Lu. "MIONet: Learning multiple-input operators via tensor product." SIAM Journal on Scientific Computing, 44(6), A3490–A3514. (2022) Paper
Fourier-MIONet Z. Jiang, M. Zhu, D. Li, Q. Li, Y. Yuan, & L. Lu. "Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration." arXiv preprint arXiv:2303.04778. (2023) Paper
PPDONet S. Mao, R. Dong, L. Lu, K. M. Yi, S. Wang, & P. Perdikaris. "PPDONet: Deep operator networks for fast prediction of steady-state solutions in disk-planet systems." The Astrophysical Journal Letters, 950(2), L12. (2023) Paper
Fourier-DeepONet M. Zhu, S. Feng, Y. Lin, & L. Lu. "Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness." arXiv preprint arXiv:2305.17289. (2023) Paper
DeepONet Lu, Lu, Pengzhan Jin, and George Em Karniadakis. "Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators." arXiv preprint arXiv:1910.03193 (2019). Paper, Video

Libraries

  • torchdyn: A PyTorch based library for all things neural differential equations.
  • DeepXDE: DeepXDE is a library for scientific machine learning and physics-informed learning, written in Python. It supports multiple deep learning backends: Tensorflow, Pytorch, Jax and PaddlePaddle.

Videos

Blogs

Research Groups

Contact

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