| [1] |
Brunton, S. L.; Kutz, J. N., Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (2022), Cambridge University Press · Zbl 1487.68001 |
| [2] |
Creswell, A.; White, T.; Dumoulin, V.; Arulkumaran, K.; Sengupta, B.; Bharath, A. A., Generative adversarial networks: An overview, IEEE Signal Process. Mag., 35, 53-65 (2018) · doi:10.1109/MSP.2017.2765202 |
| [3] |
Zhang, J.; Wang, Y.; Molino, P.; Li, L.; Ebert, D. S., Manifold: A model-agnostic framework for interpretation and diagnosis of machine learning models, IEEE Trans. Visualiz. Comp. Graphics, 25, 364-373 (2018) · doi:10.1109/TVCG.2018.2864499 |
| [4] |
Roscher, R.; Bohn, B.; Duarte, M. F.; Garcke, J., Explainable machine learning for scientific insights and discoveries, IEEE Access, 8, 42200-42216 (2020) · doi:10.1109/ACCESS.2020.2976199 |
| [5] |
Tanaka, G.; Yamane, T.; Héroux, J. B.; Nakane, R.; Kanazawa, N.; Takeda, S.; Numata, H.; Nakano, D.; Hirose, A., Recent advances in physical reservoir computing: A review, Neural Netw., 115, 100-123 (2019) · doi:10.1016/j.neunet.2019.03.005 |
| [6] |
Prokhorov, D., “Echo state networks: Appeal and challenges,” in Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005 (IEEE, 2005), Vol. 3, pp. 1463-1466. |
| [7] |
Hochreiter, S., The vanishing gradient problem during learning recurrent neural nets and problem solutions, Int. J. Uncertain. Fuzziness Knowledge-Based Syst., 6, 107-116 (1998) · Zbl 1087.68616 · doi:10.1142/S0218488598000094 |
| [8] |
Haluszczynski, A.; Aumeier, J.; Herteux, J.; Räth, C., Reducing network size and improving prediction stability of reservoir computing, Chaos, 30, 063136 (2020) · Zbl 1440.37076 · doi:10.1063/5.0006869 |
| [9] |
Holzmann, G., “Reservoir computing: A powerful black-box framework for nonlinear audio processing,” in International Conference on Digital Audio Effects (DAFx) (Citeseer, 2009). |
| [10] |
Carroll, T. L.; Pecora, L. M., Network structure effects in reservoir computers, Chaos, 29, 083130 (2019) · Zbl 1490.68105 · doi:10.1063/1.5097686 |
| [11] |
Haluszczynski, A.; Räth, C., Good and bad predictions: Assessing and improving the replication of chaotic attractors by means of reservoir computing, Chaos, 29, 103143 (2019) · Zbl 1425.37025 · doi:10.1063/1.5118725 |
| [12] |
Maass, W.; Natschläger, T.; Markram, H., Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput., 14, 2531-2560 (2002) · Zbl 1057.68618 · doi:10.1162/089976602760407955 |
| [13] |
Jaeger, H., “The “echo state” approach to analysing and training recurrent neural networks-with an erratum note,” German National Research Center for Information Technology GMD Technical Report, Bonn, Germany, No. 148, p. 13 (2001). |
| [14] |
Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139 · doi:10.1038/30918 |
| [15] |
Albert, R.; Barabási, A.-L., Statistical mechanics of complex networks, Rev. Mod. Phys., 74, 47 (2002) · Zbl 1205.82086 · doi:10.1103/RevModPhys.74.47 |
| [16] |
Broido, A. D.; Clauset, A., Scale-free networks are rare, Nat. Commun., 10, 1017 (2019) · doi:10.1038/s41467-019-08746-5 |
| [17] |
Gerlach, M.; Altmann, E. G., Testing statistical laws in complex systems, Phys. Rev. Lett., 122, 168301 (2019) · doi:10.1103/PhysRevLett.122.168301 |
| [18] |
Griffith, A.; Pomerance, A.; Gauthier, D. J., Forecasting chaotic systems with very low connectivity reservoir computers, Chaos, 29, 123108 (2019) · doi:10.1063/1.5120710 |
| [19] |
Pathak, J.; Lu, Z.; Hunt, B. R.; Girvan, M.; Ott, E., Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data, Chaos, 27, 121102 (2017) · Zbl 1390.37138 · doi:10.1063/1.5010300 |
| [20] |
Lu, Z.; Hunt, B. R.; Ott, E., Attractor reconstruction by machine learning, Chaos, 28, 061104 (2018) · doi:10.1063/1.5039508 |
| [21] |
Lorenz, E. N., Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
| [22] |
Herteux, J.; Räth, C., Breaking symmetries of the reservoir equations in echo state networks, Chaos, 30, 123142 (2020) · Zbl 1451.37109 · doi:10.1063/5.0028993 |
| [23] |
Vaidyanathan, S. and Azar, A. T., “Adaptive control and synchronization of Halvorsen circulant chaotic systems,” in Advances in Chaos Theory and Intelligent Control (Springer, 2016), pp. 225-247. · Zbl 1359.93248 |
| [24] |
Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I (1993), Springer · Zbl 0789.65048 |
| [25] |
Jaeger, H.; Haas, H., Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science, 304, 78-80 (2004) · doi:10.1126/science.1091277 |
| [26] |
Haluszczynski, A., “Prediction and control of nonlinear dynamical systems using machine learning,” Ph.D, dissertation (Ludwig-Maximilians-Universität München, 2021). |
| [27] |
Erdös, P.; Rényi, A., On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci., 5, 17-60 (1960) · Zbl 0103.16301 |
| [28] |
Hoerl, A. E.; Kennard, R. W., Ridge regression: Applications to nonorthogonal problems, Technometrics, 12, 69-82 (1970) · Zbl 0202.17206 · doi:10.1080/00401706.1970.10488635 |
| [29] |
Paige, C. C., Bidiagonalization of matrices and solution of linear equations, SIAM J. Numer. Anal., 11, 197-209 (1974) · Zbl 0244.65023 · doi:10.1137/0711019 |
| [30] |
Kobourov, S. G. |
| [31] |
Yu, D., Wang, H., Chen, P., and Wei, Z., “Mixed pooling for convolutional neural networks,” in Rough Sets and Knowledge Technology: 9th International Conference, RSKT 2014, Shanghai, China, October 24-26, 2014, Proceedings 9 (Springer, 2014), pp. 364-375. |
| [32] |
Dietterich, T. G., “Ensemble methods in machine learning,” in Multiple Classifier Systems: First International Workshop, MCS 2000 Cagliari, Italy, June 21-23, 2000 Proceedings 1 (Springer, 2000), pp. 1-15. |
| [33] |
Grassberger, P. and Procaccia, I., “Measuring the strangeness of strange attractors,” in The Theory of Chaotic Attractors (Springer, 2004), pp. 170-189. · Zbl 0593.58024 |
| [34] |
Mandelbrot, B., How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156, 636-638 (1967) · doi:10.1126/science.156.3775.636 |
| [35] |
Grassberger, P., Generalized dimensions of strange attractors, Phys. Lett. A, 97, 227-230 (1983) · doi:10.1016/0375-9601(83)90753-3 |
| [36] |
Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317 (1985) · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9 |
| [37] |
Shaw, R., Strange attractors, chaotic behavior, and information flow, Z. Naturforsch. A, 36, 80-112 (1981) · Zbl 0599.58033 · doi:10.1515/zna-1981-0115 |
| [38] |
Rosenstein, M. T.; Collins, J. J.; De Luca, C. J., A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117-134 (1993) · Zbl 0779.58030 · doi:10.1016/0167-2789(93)90009-P |
