[27] Yao Huang, Wenrui Hao, and Guang Lin. Hompinns: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations. Computers & Mathematics with Applications, 121:62-73, 2022.
[28] The MathWorks Inc. Matlab version: 9.13.0 (r2022b), 2022.
[29] Edward L Ince. Ordinary differential equations. Courier Corporation, 1956.
[30] Kenji Kawaguchi. On the theory of implicit deep learning: Global convergence with implicit layers. In International Conference on Learning Representations (ICLR), 2021.
[31] Jacob Kelly, Jesse Bettencourt, Matthew J Johnson, and David K Duvenaud. Learning differential equations that are easy to solve. Advances in Neural Information Processing Systems, 33:4370-4380, 2020.
[32] Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equations for irregular time series. Advances in Neural Information Processing Systems, 33:6696-6707, 2020.
[33] Joon-Hyuk Ko, Hankyul Koh, Nojun Park, and Wonho Jhe. Homotopy-based training of neuralodes for accurate dynamics discovery. arXiv preprint arXiv:2210.01407, 2022.
[34] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[35] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998.
[36] Takashi Matsubara, Yuto Miyatake, and Takaharu Yaguchi. Symplectic adjoint method for exact gradient of neural ode with minimal memory. Advances in Neural Information Processing Systems, 34:20772-20784, 2021.
[37] Mohammed Ali mmoustafa. Tiny imagenet, 2017.
[38] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. 2011.
[39] Jorge Nocedal and Stephen J Wright. Numerical optimization. Springer, 1999.
[40] Avik Pal, Alan Edelman, and Christopher Rackauckas. Continuous deep equilibrium models: Training neural odes faster by integrating them to infinity. arXiv preprint arXiv:2201.12240, 2022.
[41] Ashwini Pokle, Zhengyang Geng, and J Zico Kolter. Deep equilibrium approaches to diffusion models. Advances in Neural Information Processing Systems, 35:37975-37990, 2022.
[42] Lev Semenovich Pontryagin. Mathematical theory of optimal processes. CRC press, 1987.
[43] Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. Universal differential equations for scientific machine learning. arXiv preprint arXiv:2001.04385, 2020.
[44] Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equations for irregularly-sampled time series. Advances in neural information processing systems, 32, 2019.
[45] Haixiang Sun, Ye Shi, Jingya Wang, Hoang Duong Tuan, H Vincent Poor, and Dacheng Tao. Alternating differentiation for optimization layers. International Conference on Learning Representations, 2023.
[46] Ezra Winston and J Zico Kolter. Monotone operator equilibrium networks. Advances in neural information processing systems, 33:10718-10728, 2020.
[47] Xingyu Xie, Qiuhao Wang, Zenan Ling, Xia Li, Guangcan Liu, and Zhouchen Lin. Optimization induced equilibrium networks: An explicit optimization perspective for understanding equilibrium models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(3):3604-3616, 2023.
[48] Yuan Yin, Vincent Le Guen, Jérémie Dona, Emmanuel de Bézenac, Ibrahim Ayed, Nicolas Thome, and Patrick Gallinari. Augmenting physical models with deep networks for complex dynamics forecasting. Journal of Statistical Mechanics: Theory and Experiment, 2021(12):124012, 2021.