SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Conclusion By employing entropy dissipation of the MVE, we design a potential function for a hypothesis velocity field such that it controls the KL divergence between the corresponding hypothesis solution and the ground truth, for any time stamp within the period of interest. Built on this potential, we proposed the EINN method for MVEs with NN and derived the detailed computation method of the stochastic gradient, using the classic adjoint method. Through empirical studies on examples of the 2D NSE and the 3D MVE with Coulomb interactions, we show the significant advantage of the proposed method, when compared with two SOTA NN based MVE solvers.
Zhenfu Wang is supported by the National Key R&D Program of China, Project Number 2021YFA1002800, NSFC grant No.12171009, Young Elite Scientist Sponsorship Program by China Association for Science and Technology (CAST) No. YESS20200028 and the Fundamental Research Funds for the Central Universities (the start-up fund), Peking University. Zebang Shen's work is supported by ETH research grant and Swiss National Science Foundation (SNSF) Project Funding No. 200021-207343.
N. M. Boffi and E. Vanden-Eijnden. Probability flow solution of the fokker-planck equation. Machine Learning: Science and Technology, 4(3):035012, 2023.
D. Bresch, P.-E. Jabin, and Z. Wang. On mean-field limits and quantitative estimates with a large class of singular kernels: application to the patlak-keller-segel model. Comptes Rendus Mathematique, 357(9):708-720, 2019a.
D. Bresch, P.-E. Jabin, and Z. Wang. Modulated free energy and mean field limit. Séminaire Laurent Schwartz—EDP et applications, pages 1–22, 2019b.
S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis. Physics-informed neural networks (pinns) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12):1727-1738, 2021.
J. A. Carrillo, K. Craig, and F. S. Patachini. A blob method for diffusion. *Calculus of Variations and Partial Differential Equations*, 58:1-53, 2019.
J. A. Carrillo, K. Craig, L. Wang, and C. Wei. Primal dual methods for wasserstein gradient flows. Foundations of Computational Mathematics, pages 1-55, 2022.
S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli. Scientific machine learning through physics-informed neural networks: where we are and what's next. Journal of Scientific Computing, 92(3):88, 2022.
T. De Ryck and S. Mishra. Error analysis for physics-informed neural networks (pinns) approximating kolmogorov pdes. Advances in Computational Mathematics, 48(6):1-40, 2022.
T. De Ryck, S. Lanthaler, and S. Mishra. On the approximation of functions by tanh neural networks. Neural Networks, 143:732-750, 2021. ISSN 0893-6080. doi: https://doi.org/10.1016/j.neunet.2021.08.015. URL https://www.sciencedirect.com/science/article/pii/S0893608021003208
T. De Ryck, A. Jagtap, and S. Mishra. Error estimates for physics-informed neural networks approximating the navier-stokes equations. IMA Journal of Numerical Analysis, 2023.
L. Erdős and H.-T. Yau. A dynamical approach to random matrix theory, volume 28. American Mathematical Soc., 2017.
L. C. Evans. Partial differential equations, volume 19. American Mathematical Society, 2022.
F. Golse. On the dynamics of large particle systems in the mean field limit. *Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity*, pages 1-144, 2016.
A. Guillin, P. L. Bris, and P. Monmarché. Uniform in time propagation of chaos for the 2d vortex model and other singular stochastic systems. arXiv preprint arXiv:2108.08675, 2021.
G. Gupta, X. Xiao, and P. Bogdan. Multiwavelet-based operator learning for differential equations. Advances in neural information processing systems, 34:24048-24062, 2021.
J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505-8510, 2018.
P.-E. Jabin and Z. Wang. Quantitative estimates of propagation of chaos for stochastic systems with  $W^{-1,\infty}$  kernels. Inventiones mathematicae, 214(1):523-591, 2018.
Z. Shen, Z. Wang, S. Kale, A. Ribeiro, A. Karbasi, and H. Hassani. Self-consistency of the fokker planck equation. In P.-L. Loh and M. Raginsky, editors, Proceedings of Thirty Fifth Conference on Learning Theory, volume 178 of Proceedings of Machine Learning Research, pages 817-841. PMLR, 02-05 Jul 2022. URL https://proceedings.mlr.press/v178/shen22a.html
J. Shin, YeonjongDarbon and G. Em Karniadakis. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type pdes. Communications in Computational Physics, 28(5):2042-2074, 2020.
Y. Shin, Z. Zhang, and G. E. Karniadakis. Error estimates of residual minimization using neural networks for linear pdes. arXiv preprint arXiv:2010.08019, 2020.
G. D. Smith, G. D. Smith, and G. D. S. Smith. Numerical solution of partial differential equations: finite difference methods. Oxford university press, 1985.
C. Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
C. Wang, S. Li, D. He, and L. Wang. Is  $\mathcal{L}^2$  physics informed loss always suitable for training physics informed neural network? In Advances in Neural Information Processing Systems, 2022.
E. Weinan, J. Han, and A. Jentzen. Algorithms for solving high dimensional pdes: from nonlinear monte carlo to machine learning. Nonlinearity, 35(1):278, 2021.
X. Xiao, D. Cao, R. Yang, G. Gupta, G. Liu, C. Yin, R. Balan, and P. Bogdan. Coupled multiwavelet neural operator learning for coupled partial differential equations. arXiv preprint arXiv:2303.02304, 2023.
L. Zhang, J. Han, H. Wang, and R. Car. Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Physical review letters, 120(14):143001, 2018.
R. Zhang, P. Hu, Q. Meng, Y. Wang, R. Zhu, B. Chen, Z.-M. Ma, and T.-Y. Liu. Drvn (deep random vortex network): A new physics-informed machine learning method for simulating and inferring incompressible fluid flows. Physics of Fluids, 34(10):107112, 2022.