1. 毕业设计（论文）的内容和要求
近年来，机器学习的迅猛发展使得我没可以从大量观测数据中发现传统的物理方程模型和方程求解。
目前常见的机器学习方法主要有符号回归方法 [1] ，高斯过程学习方法 [2] 及采用稀疏优化的相关理论和算法[3]-[7]，传统机器学习方法发现和求解非线性微分方程存在各种不足[10]：如需要对观测数据进行光滑处理以便较稳定的数值求导[3]，先验模型过于理想化等[2]。
本课题基于深度神经网络的万有逼近性质 ，将物理方程的求解和发现归结为一个从观测噪声数据估计神经网络参数的过程，借助于自动微分技术和随机梯度下降算法 [11] ，深度网络能有效的发现和求解微分方程模型。
2. 参考文献
[1]. Schmidt M, Lipson H. 2009 Distilling free-form natural laws from experimental data. Science 324, 8185. [2]. Raissi, Maziar, Perdikaris, Paris, and Karniadakis, George Em. Machine Learning of Linear Differential Equations using Gaussian Processes, Journal of Computational Physics 2017[URL] [3]. Brunton SL, Proctor JL, Kutz JN. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 39323937. [4]. Hayden Schaeffer. Learning partial differential equations via data discovery and sparse opti-mization. Proc. R. Soc. A, 473(2197):20160446, 2017 [5]. Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Data-driven discovery of partial differential equations. Science Advances, 3(4): e1602614, 2017 [6]. Samuel Rudy, Alessandro Alla, Steven L. Brunton, J. Nathan Kutz. Data-driven identification of parametric partial differential equations. arXiv:1806.00732 [math.NA] [7]. Giang Tran and Rachel Ward. Exact recovery of chaotic systems from highly corrupted data. Multiscale Modeling lm Gunes Baydin et.al. Automatic Differentiation in Machine Learning: a Survey. Journal of Machine Learning Research, 18(153):1--43, 2018.[12]. TongQin, KailiangWu, DongbinXiu.Data driven governing equations approximation using deep neural networks. Journal of Computational Physics 2019 620-635.[13]. Samuel H.Rudy,J.NathanKutz, Steven L.Brunton.Deep learning of dynamics and signal-noise decomposition with time-stepping constraints Journal of Computational Physics 2019 483-506.[14]. Alexandre M. Tartakovsky et.al. Learning Parameters and Constitutive Relationshipswith Physics Informed Deep Neural Networks. arXiv:1808.03398v2
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