- [1]
E. Hairer, C. Lubich and G. Wanner.
Geometric Numerical integration: structure-preserving algorithms for ordinary differential equations (Springer, Heidelberg, 2006).
↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10 ↩11 ↩12- [2]
- [3]
B. Brantner and M. Kraus.
Symplectic autoencoders for Model Reduction of Hamiltonian Systems, arXiv preprint arXiv:2312.10004 (2023).
↩1 ↩2 ↩3 ↩4- [4]
B. Brantner, M. Kraus, Z. Li and G. de Romemont.
Volume-preserving transformers for learning time series data with structure. ESAIM: Proceedings and Surveys
81, 123–144 (2025).
↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10 ↩11- [5]
P. Jin, Z. Zhang, A. Zhu, Y. Tang and G. E. Karniadakis.
SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Neural Networks
132, 166–179 (2020).
↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10 ↩11- [6]
S. Greydanus, M. Dzamba and J. Yosinski.
Hamiltonian neural networks. Advances in neural information processing systems
32 (2019).
↩1 ↩2 ↩3 ↩4- [7]
B. Brantner.
Generalizing Adam To Manifolds For Efficiently Training Transformers, arXiv preprint arXiv:2305.16901 (2023).
↩1 ↩2 ↩3 ↩4- [8]
L. Kong, Y. Wang and M. Tao.
Momentum stiefel optimizer, with applications to suitably-orthogonal attention, and optimal transport, arXiv preprint arXiv:2205.14173v3 (2023).
↩1 ↩2 ↩3- [9]
A. Zhang, A. Chan, Y. Tay, J. Fu, S. Wang, S. Zhang, H. Shao, S. Yao and R. K.-W. Lee.
On orthogonality constraints for transformers. In:
Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Vol. 2 (Association for Computational Linguistics, 2021); pp. 375–382.
↩1 ↩2- [10]
- [11]
- [12]
- [13]
- [14]
S. I. Richard L. Bishop.
Tensor Analysis on Manifolds (Dover Publications, Mineola, New York, 1980).
↩1 ↩2 ↩3- [15]
W. S. Moses, V. Churavy, L. Paehler, J. Hückelheim, S. H. Narayanan, M. Schanen and J. Doerfert.
Reverse-Mode Automatic Differentiation and Optimization of GPU Kernels via Enzyme. In:
Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis,
SC '21 (Association for Computing Machinery, New York, NY, USA, 2021).
↩1- [16]
M. Betancourt.
A geometric theory of higher-order automatic differentiation, arXiv preprint arXiv:1812.11592 (2018).
↩1- [17]
J. Bolte and E. Pauwels.
A mathematical model for automatic differentiation in machine learning. Advances in Neural Information Processing Systems
33, 10809–10819 (2020).
↩1- [18]
V. I. Arnold.
Mathematical methods of classical mechanics. Vol. 60 of
Graduate Texts in Mathematics (Springer Verlag, Berlin, 1978).
↩1 ↩2 ↩3 ↩4- [19]
M. Kraus, K. Kormann, P. J. Morrison and E. Sonnendrücker.
GEMPIC: geometric electromagnetic particle-in-cell methods. Journal of Plasma Physics
83, 905830401 (2017).
↩1- [20]
Z. Ge and J. E. Marsden.
Lie-poisson hamilton-jacobi theory and lie-poisson integrators. Physics Letters A
133, 134–139 (1988).
↩1- [21]
S. Lang.
Fundamentals of differential geometry. Vol. 191 (Springer Science & Business Media, 2012).
↩1- [22]
K. Hornik, M. Stinchcombe and H. White.
Multilayer feedforward networks are universal approximators. Neural networks
2, 359–366 (1989).
↩1 ↩2- [23]
- [24]
D.-X. Zhou. Universality of deep convolutional neural networks. Applied and computational harmonic analysis 48, 787–794 (2020).
- [25]
Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou and M. Tegmark.
Kan: Kolmogorov-arnold networks, arXiv preprint arXiv:2404.19756 (2024).
↩1- [26]
J. W. Burby, Q. Tang and R. Maulik.
Fast neural Poincaré maps for toroidal magnetic fields. Plasma Physics and Controlled Fusion
63, 024001 (2020).
↩1- [27]
P. Horn, V. S. Ulibarrena, B. Koren and S. P. Zwart.
A generalized framework of neural networks for Hamiltonian systems. Journal of Computational Physics
521, 113536 (2025).
↩1 ↩2 ↩3- [28]
E. Celledoni, M. J. Ehrhardt, C. Etmann, R. I. McLachlan, B. Owren, C.-B. Schonlieb and F. Sherry.
Structure-preserving deep learning. European journal of applied mathematics
32, 888–936 (2021).
↩1- [29]
J. Bajārs.
Locally-symplectic neural networks for learning volume-preserving dynamics. Journal of Computational Physics
476, 111911 (2023).
↩1 ↩2- [30]
F. Kang and S. Zai-Jiu.
Volume-preserving algorithms for source-free dynamical systems. Numerische Mathematik
71, 451–463 (1995).
↩1- [31]
H. Cardot.
Recurrent neural networks for temporal data processing (BoD–Books on Demand, 2011).
↩1- [32]
D. Bahdanau, K. Cho and Y. Bengio.
Neural machine translation by jointly learning to align and translate, arXiv preprint arXiv:1409.0473 (2014).
↩1 ↩2- [33]
A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser and I. Polosukhin.
Attention is all you need. Advances in neural information processing systems
30 (2017).
↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10- [34]
K. Jacobs.
Discrete Stochastics (Birkhäuser Verlag, Basel, Switzerland, 1992).
↩1- [35]
- [36]
M.-T. Luong, H. Pham and C. D. Manning.
Effective approaches to attention-based neural machine translation, arXiv preprint arXiv:1508.04025 (2015).
↩1- [37]
K. Feng and M.-z. Qin.
The symplectic methods for the computation of Hamiltonian equations. In:
Numerical Methods for Partial Differential Equations: Proceedings of a Conference held in Shanghai, PR China, March 25–29, 1987 (Springer, 1987); pp. 1–37.
↩1 ↩2- [38]
Z. Ge and K. Feng.
On the approximation of linear Hamiltonian systems. Journal of Computational Mathematics, 88–97 (1988).
↩1 ↩2- [39]
- [40]
T. Blickhan.
A registration method for reduced basis problems using linear optimal transport, arXiv preprint arXiv:2304.14884 (2023).
↩1 ↩2 ↩3- [41]
S. Fresca, L. Dede’ and A. Manzoni.
A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. Journal of Scientific Computing
87, 1–36 (2021).
↩1 ↩2- [42]
K. Lee and K. T. Carlberg.
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. Journal of Computational Physics
404, 108973 (2020).
↩1 ↩2 ↩3 ↩4- [43]
M. J. Gander and G. Wanner.
From Euler, Ritz, and Galerkin to modern computing. Siam Review
54, 627–666 (2012).
↩1- [44]
F. Arbes, C. Greif and K. Urban.
The Kolmogorov N-width for linear transport: Exact representation and the influence of the data, arXiv preprint arXiv:2305.00066 (2023).
↩1- [45]
C. Greif and K. Urban.
Decay of the Kolmogorov N-width for wave problems. Applied Mathematics Letters
96, 216–222 (2019).
↩1- [46]
A. Chatterjee.
An introduction to the proper orthogonal decomposition. Current science, 808–817 (2000).
↩1- [47]
S. Volkwein.
Proper orthogonal decomposition: Theory and reduced-order modelling. Lecture Notes, University of Konstanz
4, 1–29 (2013).
↩1- [48]
I. Goodfellow, Y. Bengio and A. Courville.
Deep learning (MIT press, Cambridge, MA, 2016).
↩1 ↩2 ↩3 ↩4- [49]
L. Peng and K. Mohseni.
Symplectic model reduction of Hamiltonian systems. SIAM Journal on Scientific Computing
38, A1–A27 (2016).
↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8- [50]
T. M. Tyranowski and M. Kraus.
Symplectic model reduction methods for the Vlasov equation. Contributions to Plasma Physics
63, e202200046 (2023).
↩1- [51]
P. Buchfink, S. Glas and B. Haasdonk.
Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder. SIAM Journal on Scientific Computing
45, A289–A311 (2023).
↩1 ↩2 ↩3 ↩4- [52]
M. Raissi, P. Perdikaris and G. 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, 686–707 (2019).
↩1 ↩2 ↩3 ↩4 ↩5- [53]
S. Yıldız, P. Goyal, T. Bendokat and P. Benner.
Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems Using Weakly Symplectic Liftings. Journal of Machine Learning for Modeling and Computing
5 (2024).
↩1 ↩2 ↩3- [54]
A. Van Der Schaft, D. Jeltsema and others.
Port-Hamiltonian systems theory: An introductory overview. Foundations and Trends in Systems and Control
1, 173–378 (2014).
↩1 ↩2- [55]
R. Morandin.
Modeling and numerical treatment of port-Hamiltonian descriptor systems. Ph.D. Thesis, Technische Universität Berlin (2023).
↩1- [56]
H. Yoshimura and J. E. Marsden.
Dirac structures in Lagrangian mechanics Part I: implicit Lagrangian systems. Journal of Geometry and Physics
57, 133–156 (2006).
↩1- [57]
H. Yoshimura and J. E. Marsden.
Dirac structures in Lagrangian mechanics Part II: Variational structures. Journal of Geometry and Physics
57, 209–250 (2006).
↩1- [58]
P. Kotyczka and L. Lefevre.
Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters
133, 104530 (2019).
↩1 ↩2- [59]
V. Mehrmann and R. Morandin.
Structure-preserving discretization for port-Hamiltonian descriptor systems. In:
2019 IEEE 58th Conference on Decision and Control (CDC) (IEEE, 2019); pp. 6863–6868.
↩1- [60]
T. F. Moser.
Structure-Preserving Model Reduction of Port-Hamiltonian Descriptor Systems. Ph.D. Thesis, Technische Universität München (2023).
↩1 ↩2- [61]
J. Rettberg, J. Kneifl, J. Herb, P. Buchfink, J. Fehr and B. Haasdonk.
Data-driven identification of latent port-Hamiltonian systems, arXiv preprint arXiv:2408.08185 (2024).
↩1- [62]
S. E. Otto, G. R. Macchio and C. W. Rowley.
Learning nonlinear projections for reduced-order modeling of dynamical systems using constrained autoencoders. Chaos: An Interdisciplinary Journal of Nonlinear Science
33 (2023).
↩1- [63]
B. Leimkuhler and S. Reich.
Simulating hamiltonian dynamics. No. 14 (Cambridge university press, 2004).
↩1 ↩2- [64]
S. Hochreiter and J. Schmidhuber.
Long short-term memory. Neural computation
9, 1735–1780 (1997).
↩1 ↩2- [65]
A. Hemmasian and A. Barati Farimani.
Reduced-order modeling of fluid flows with transformers. Physics of Fluids
35 (2023).
↩1- [66]
A. Solera-Rico, C. S. Vila, M. Gómez, Y. Wang, A. Almashjary, S. Dawson and R. Vinuesa,
$\beta$-Variational autoencoders and transformers for reduced-order modelling of fluid flows, arXiv preprint arXiv:2304.03571 (2023).
↩1 ↩2 ↩3- [67]
P. Jin, Z. Lin and B. Xiao.
Optimal unit triangular factorization of symplectic matrices. Linear Algebra and its Applications (2022).
↩1- [68]
- [69]
A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly and others.
An image is worth 16x16 words: Transformers for image recognition at scale, arXiv preprint arXiv:2010.11929 (2020).
↩1 ↩2 ↩3 ↩4- [70]
M. Toda.
Vibration of a chain with nonlinear interaction. Journal of the Physical Society of Japan
22, 431–436 (1967).
↩1 ↩2- [71]
C. Villani and others.
Optimal transport: old and new. Vol. 338 (Springer, 2009).
↩1- [72]
C. Villani.
Topics in optimal transportation. Vol. 58 (American Mathematical Soc., 2021).
↩1- [73]
- [74]
- [75]
J. Achiam, S. Adler, S. Agarwal, L. Ahmad, I. Akkaya, F. L. Aleman, D. Almeida, J. Altenschmidt, S. Altman, S. Anadkat and others.
Gpt-4 technical report, arXiv preprint arXiv:2303.08774 (2023).
↩1 ↩2- [76]
Y. Duan, J. S. Edwards and Y. K. Dwivedi.
Artificial intelligence for decision making in the era of Big Data–evolution, challenges and research agenda. International journal of information management
48, 63–71 (2019).
↩1- [77]
D. C. Psichogios and L. H. Ungar.
A hybrid neural network-first principles approach to process modeling. AIChE Journal
38, 1499–1511 (1992).
↩1- [78]
N. Baker, F. Alexander, T. Bremer, A. Hagberg, Y. Kevrekidis, H. Najm, M. Parashar, A. Patra, J. Sethian, S. Wild and others.
Workshop report on basic research needs for scientific machine learning: Core technologies for artificial intelligence (USDOE Office of Science (SC), Washington, DC (United States), 2019).
↩1- [79]
D. N. Arnold, R. S. Falk and R. Winther.
Finite element exterior calculus, homological techniques, and applications. Acta numerica
15, 1–155 (2006).
↩1- [80]
Y. Lishkova, P. Scherer, S. Ridderbusch, M. Jamnik, P. Liò, S. Ober-Blöbaum and C. Offen.
Discrete Lagrangian neural networks with automatic symmetry discovery. IFAC-PapersOnLine
56, 3203–3210 (2023).
↩1- [81]
E. Dierkes, C. Offen, S. Ober-Blöbaum and K. Flaßkamp.
Hamiltonian neural networks with automatic symmetry detection. Chaos: An Interdisciplinary Journal of Nonlinear Science
33 (2023).
↩1- [82]
- [83]
- [84]
- [85]
- [86]
- [87]
D. Bon, G. Pai, G. Bellaard, O. Mula and R. Duits.
Optimal Transport on the Lie Group of Roto-translations, arXiv preprint arXiv:2402.15322 (2024).
↩1- [88]
T. Bendokat, R. Zimmermann and P.-A. Absil.
A Grassmann manifold handbook: Basic geometry and computational aspects, arXiv preprint arXiv:2011.13699 (2020).
↩1- [89]
T. Bendokat and R. Zimmermann.
The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications, arXiv preprint arXiv:2108.12447 (2021).
↩1- [90]
D. Kingma.
Adam: a method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014).
↩1 ↩2- [91]
P.-A. Absil, R. Mahony and R. Sepulchre.
Optimization algorithms on matrix manifolds (Princeton University Press, Princeton, New Jersey, 2008).
↩1- [92]
L. Deng.
The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine
29, 141–142 (2012).
↩1 ↩2- [93]
T. Frankel.
The geometry of physics: an introduction (Cambridge university press, Cambridge, UK, 2011).
↩1- [94]
J. Li, L. Fuxin and S. Todorovic.
Efficient riemannian optimization on the stiefel manifold via the cayley transform, arXiv preprint arXiv:2002.01113 (2020).
↩1- [95]
L. Huang, X. Liu, B. Lang, A. Yu, Y. Wang and B. Li.
Orthogonal weight normalization: Solution to optimization over multiple dependent stiefel manifolds in deep neural networks. In:
Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 32 no. 1 (2018).
↩1- [96]
R. Xiong, Y. Yang, D. He, K. Zheng, S. Zheng, C. Xing, H. Zhang, Y. Lan, L. Wang and T. Liu.
On layer normalization in the transformer architecture. In:
International Conference on Machine Learning (PMLR, 2020); pp. 10524–10533.
↩1- [97]
S. Fresca and A. Manzoni.
POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Computer Methods in Applied Mechanics and Engineering
388, 114181 (2022).
↩1- [98]
P. J. Morrison.
A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena
18, 410–419 (1986).
↩1- [99]
A. Gruber, M. Gunzburger, L. Ju and Z. Wang.
Energetically consistent model reduction for metriplectic systems. Computer Methods in Applied Mechanics and Engineering
404, 115709 (2023).
↩1- [100]
P. Schulze. Structure-preserving model reduction for port-hamiltonian systems based on a special class of nonlinear approximation ansatzes, arXiv preprint arXiv:2302.06479 (2023).
- [101]
M. Mamunuzzaman and H. Zwart.
Structure preserving model order reduction of port-Hamiltonian systems, arXiv preprint arXiv:2203.07751 (2022).
↩1- [102]
S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth.
Hybrid monte carlo. Physics letters B
195, 216–222 (1987).
↩1- [103]
A. D. Cobb and B. Jalaian.
Scaling Hamiltonian Monte Carlo inference for Bayesian neural networks with symmetric splitting. In:
Uncertainty in Artificial Intelligence (PMLR, 2021); pp. 675–685.
↩1- [104]
A. Fichtner and S. Simutė.
Hamiltonian Monte Carlo inversion of seismic sources in complex media. Journal of Geophysical Research: Solid Earth
123, 2984–2999 (2018).
↩1- [105]
A. Wibisono, A. C. Wilson and M. I. Jordan.
A variational perspective on accelerated methods in optimization, proceedings of the National Academy of Sciences
113, E7351–E7358 (2016).
↩1- [106]
V. Duruisseaux and M. Leok.
Accelerated optimization on Riemannian manifolds via discrete constrained variational integrators. Journal of Nonlinear Science
32, 42 (2022).
↩1- [107]
H. Sato and K. Aihara.
Cholesky QR-based retraction on the generalized Stiefel manifold. Computational Optimization and Applications
72, 293–308 (2019).
↩1- [108]
B. Gao, N. T. Son and T. Stykel.
Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications. Linear Algebra and its Applications
682, 50–85 (2024).
↩1- [109]
K. Feng. The step-transition operators for multi-step methods of ODE's. Journal of Computational Mathematics, 193–202 (1998).
- [110]
P.-A. Absil, R. Mahony and R. Sepulchre. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica 80, 199–220 (2004).
- [111]
B. Brantner, G. de Romemont, M. Kraus and Z. Li. Structure-Preserving Transformers for Learning Parametrized Hamiltonian Systems, arXiv preprint arXiv:2312:11166 (2023).
- [112]
E. Celledoni and A. Iserles. Approximating the exponential from a Lie algebra to a Lie group. Mathematics of Computation 69, 1457–1480 (2000).
- [113]
M. P. Do Carmo and J. Flaherty Francis. Riemannian geometry. Vol. 2 (Springer, 1992).
- [114]
C. Fraikin, K. Hüper and P. V. Dooren. Optimization over the Stiefel manifold. In: PAMM: Proceedings in Applied Mathematics and Mechanics, Vol. 7 no. 1 (Wiley Online Library, 2007); pp. 1062205–1062206.
- [115]
B. Gao, N. T. Son, P.-A. Absil and T. Stykel. Riemannian optimization on the symplectic Stiefel manifold. SIAM Journal on Optimization 31, 1546–1575 (2021).
- [116]
B. Gao, N. T. Son and T. Stykel. Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton's methods, arXiv preprint arXiv:2406.14299 (2024).
- [117]
D. D. Holm, T. Schmah and C. Stoica. Geometric mechanics and symmetry: from finite to infinite dimensions. Vol. 12 (Oxford University Press, Oxford, UK, 2009).
- [118]
W. Huang, P.-A. Absil and K. A. Gallivan. A Riemannian BFGS method for nonconvex optimization problems. In: Numerical Mathematics and Advanced Applications ENUMATH 2015 (Springer, 2016); pp. 627–634.
- [119]
P. Kenneweg, T. Kenneweg and B. Hammer. Improving Line Search Methods for Large Scale Neural Network Training. In: 2024 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA) (IEEE, 2024); pp. 1–6.
- [120]
S. Lang. Real and functional analysis. Vol. 142 (Springer Science & Business Media, 2012).
- [121]
T. Lin and H. Zha. Riemannian manifold learning. IEEE transactions on pattern analysis and machine intelligence 30, 796–809 (2008).
- [122]
S. Lipschutz. General Topology (McGraw-Hill Book Company, New York City, New York, 1965).
- [123]
F. Mezzadri. How to generate random matrices from the classical compact groups, arXiv preprint math-ph/0609050 (2006).
- [124]
B. O'neill. Semi-Riemannian geometry with applications to relativity (Academic press, New York City, New York, 1983).
- [125]
M. Schlarb. Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport. The Journal of Geometric Analysis 34, 1–43 (2024).
- [126]
J. N. Stephen J. Wright. Numerical optimization (Springer Science+Business Media, New York, NY, 2006).
- [127]
S. Vaswani, A. Mishkin, I. Laradji, M. Schmidt, G. Gidel and S. Lacoste-Julien. Painless stochastic gradient: Interpolation, line-search, and convergence rates. Advances in neural information processing systems 32 (2019).
- [128]