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 | 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 | ||
| 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 |
- 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.
- 2020-10-16 - Jaideep Pathak - Using ML to Augment Coarse-Grid CFD Simulations
- Steve Brunton: Machine Learning for Fluid Dynamics
- Petros Koumoutsakos: "Machine Learning for Fluid Mechanics"
- Brunton Lab: Data-driven dynamics and control
- Animashree Anandkumar
- Wing Kam Liu Group
- George M. Karniadakis
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