[人工智能]内嵌物理知识深度学习方法与不确定性量化论文总结

内嵌物理知识深度学习方法(PINN)与不确定性量化论文总结

方法综述

量化预测的不确定性，如来自噪声数据和未知项的不确定性，对于基于神经网络的预测模型具有重要意义。 然而，内嵌物理知识神经网络中没有内置不确定度量化(UQ)，这可能会限制一些需要高可靠性的应用。 对于一些UQ问题，生成输出预测的分布更为重要比直接获得解的估计值重要。 对此，Yang等[1,2]对通过结合生成对抗网络与基于物理的损失函数对系统中不确定性进行了量化，其中，隐藏层变量被构造为概率表示，然后对模型进行训练。 而系统的物理信息编码到损失函数而不是鉴别器，没有充分挖掘adversarial inference的潜力。由此，Daw等人[3]提出了一个基于物理信息鉴别器的GAN框架，其中物理定律被编码到两个生成器和鉴别器模型中。 此外，在深度学习中贝叶斯方法也是量化UQ的重要方法[4,5,6]。 Yang等人[7]结合了贝叶斯神经网络和内嵌物理知识神经网络能够进行预测同时量化出给定的噪声数据下的不确定性，其中，将先验概率分布引入在网络权重中，然后采用变分inference对权重近似inference，但引入贝叶斯方法带来了更多的参数和更多的计算成本。 为此，dropout被用来估计[8]的不确定性，其中任意将多项式混沌与内嵌物理神经网络相结合，求解随机偏微分方程问题。 在目前的研究，主要集中在将GAN、贝叶斯方法法或dropout与内嵌物理神经网络相结合，量化预测不确定性。

[1] Y. Yang, P. Perdikaris, Adversarial uncertainty quantification in physics-informed neural networks, Journal of Computational Physics 394 (2019) 136–152.
[2] L. Yang, D. Zhang, G. E. Karniadakis, Physics-informed generative adversarial networks for stochastic differential equations, SIAM Journal on Scientific Computing 42 (1) (2020) A292–A317.
[3] A. Daw, M. Maruf, A. Karpatne, Pid-gan: A gan framework based on a physics-informed discriminator for uncertainty quantification with physics, in: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, 2021, pp. 237–247.
[4] L. V. Jospin, W. Buntine, F. Boussaid, H. Laga, M. Bennamoun, Hands-on bayesian neural networks–a tutorial for deep learning users, arXiv preprint arXiv:2007.06823.
[5] E. Goan, C. Fookes, Bayesian neural networks: An introduction and survey, in: Case Studies in Applied Bayesian Data Science, Springer, 2020, pp. 45–87.
[6] A. G. Wilson, P. Izmailov, Bayesian deep learning and a probabilistic perspective of generalization, Advances in neural information processing systems 33 (2020) 4697–4708.
[7] L. Yang, X. Meng, G. E. Karniadakis, B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data, Journal of Computational Physics 425 (2021) 109913.
[8] D. Zhang, L. Lu, L. Guo, G. E. Karniadakis, Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems, Journal of Computational Physics 397 (2019) 108850.

不确定性与内嵌物理深度学习方法结合论文清单

• Psaros A F, Meng X, Zou Z, et al. Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons[J]. arXiv preprint arXiv:2201.07766, 2022.
  综述论文
• Zhang D, Lu L, Guo L, et al. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems[J]. Journal of Computational Physics, 2019, 397: 108850.
• Zhu Y, Zabaras N, Koutsourelakis P S, et al. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data[J]. Journal of Computational Physics, 2019, 394: 56-81.
• Yang L, Meng X, Karniadakis G E. B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data[J]. Journal of Computational Physics, 2021, 425: 109913.
• Gao Y, Ng M K. Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks[J]. Journal of Computational Physics, 2022: 111270.
• Yang L, Zhang D, Karniadakis G E. Physics-informed generative adversarial networks for stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2020, 42(1): A292-A317.
• Molnar J P, Grauer S J. Flow field tomography with uncertainty quanti?cation using a Bayesian physics-informed neural network[J]. Measurement Science and Technology, 2022.
• Daw A, Maruf M, Karpatne A. PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics[C]//Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining. 2021: 237-247.
• Yang Y, Perdikaris P. Adversarial uncertainty quantification in physics-informed neural networks[J]. Journal of Computational Physics, 2019, 394: 136-152.