Anderson, PW Plus is different. Science 177393–396 (1972).
Thompson, JMT & Stewart, HB Nonlinear dynamics and chaos (Wiley, 2002).
Hirsch, MW, Smale, S. & Devaney, RL Differential equations, dynamical systems and introduction to chaos (Academic, 2012).
Kutz, JN, Brunton, SL, Brunton, BW and Proctor, JL Dynamic mode decomposition: data-driven modeling of complex systems (SIAM, 2016).
Evans, J. & Rzhetsky, A. Machine Science. Science 329399-400 (2010).
Fortunato, S. et al. Science of science. Science 359eaao0185 (2018).
Bongard, J. & Lipson, H. Automated Reverse Engineering of Nonlinear Dynamical Systems. proc. Natl Acad. Science. UNITED STATES 1049943–9948 (2007).
Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 32481–85 (2009).
King, RD, Muggleton, SH, Srinivasan, A. & Sternberg, M. Structure-activity relationships derived from machine learning: the use of atoms and their bonding connectivities to predict mutagenicity through inductive logic programming. proc. Natl Acad. Science. UNITED STATES 93438–442 (1996).
Waltz, D. & Buchanan, BG Automation Science. Science 32443–44 (2009).
King, RD et al. The robot scientist Adam. computer 4246–54 (2009).
Langley, P. BACON: a production system that discovers empirical laws. In proc. Fifth International Joint Conference on Artificial Intelligence Flight. 1344 (Morgan Kaufmann, 1977).
Langley, P. Rediscovering Physics with BACON.3. In proc. Sixth International Joint Conference on Artificial Intelligence Flight. 1 505–507 (Morgan Kaufmann, 1979).
Crutchfield, JP & McNamara, B. Equations of Motion from a Data Series. Complex system. 1417–452 (1987).
Kevrekidis, IG et al. Equation-free and coarse-grained multi-scale computing: allowing microscopic simulators to perform system-level analysis. Common. Math. Science. 1715–762 (2003).
Yao, C. & Bollt, EM Nonlinear Parameter Modeling and Estimation with Kronecker Product Representation for Coupled Oscillators and Spatiotemporal Systems. Physics D 22778–99 (2007).
Rowley, CW, Mezić, I., Bagheri, S., Schlatter, P. & Henningson, DS Spectral analysis of nonlinear flows. J. Fluid Mech. 641115–127 (2009).
Schmidt, MD et al. Automated refinement and inference of analytical models for metabolic networks. Phys. Biol. 8055011 (2011).
Sugihara, G. et al. Detecting causality in complex ecosystems. Science 338496–500 (2012).
Ye, H. et al. Equation-free mechanistic ecosystem prediction using empirical dynamic modeling. proc. Natl Acad. Science. UNITED STATES 112E1569–E1576 (2015).
Google Scholar
Daniels, BC & Nemenman, I. Automated adaptive inference of dynamic phenomenological models. Nat. Common. 68133 (2015).
Daniels, BC & Nemenman, I. Efficient inference of parsimonious phenomenological models of cell dynamics using S-systems and alternating regression. Plos ONE tene0119821 (2015).
Benner, P., Gugercin, S. & Willcox, K. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57483–531 (2015).
Brunton, SL, Proctor, JL & Kutz, JN Discovery of governing equations from data by parsimonious identification of nonlinear dynamical systems. proc. Natl Acad. Science. UNITED STATES 1133932–3937 (2016).
Rudy, SH, Brunton, SL, Proctor, JL & Kutz, JN Discovery based on data from partial differential equations. Science. Adv. 3e1602614 (2017).
Udrescu, S.-M. & Tegmark, M. AI Feynman: A Physics-Inspired Method for Symbolic Regression. Science. Adv. 6eaay2631 (2020).
Mrowca D et al. Flexible neural representation for physical prediction. In Advances in Neural Information Processing Systems Flight. 31 (eds Bengio, S. et al.) (Curran Associates, 2018).
Champion, K., Lusch, B., Kutz, JN & Brunton, SL Discovery based on coordinate data and governing equations. proc. Natl Acad. Science. UNITED STATES 11622445–22451 (2019).
Baldi, P. & Hornik, K. Neural networks and principal component analysis: learning from examples without local minima. Neural network. 253–58 (1989).
Hinton, GE & Zemel, RS autoencoders, minimum description length and Helmholtz free energy. Adv. Neural information. Treat. System 63 (1994).
Google Scholar
Masci, J., Meier, U., Cireşan, D. & Schmidhuber, J. Stacked convolutional autoencoders for hierarchical feature extraction. In International Conference on Artificial Neural Networks 52–59 (Springer, 2011).
Bishop CM et al. Neural networks for pattern recognition (Oxford Univ. Press, 1995).
Camastra, F. & Staiano, A. Estimation of the intrinsic dimension: advances and open problems. Inf. Science. 32826–41 (2016).
Campadelli, P., Casiraghi, E., Ceruti, C. & Rozza, A. Intrinsic dimension estimation: relevant techniques and frame of reference. Math. Problem. Eng. 2015759567 (2015).
Levina, E. & Bickel, PJ Maximum likelihood estimation of intrinsic dimension. In proc. 17th International Conference on Neural Information Processing Systems 777–784 (MIT Press, 2005).
Rozza, A., Lombardi, G., Ceruti, C., Casiraghi, E. & Campadelli, P. New High Intrinsic Dimensionality Estimators. Mach. Learn. 8937–65 (2012).
Ceruti, C. et al. DANCo: an intrinsic dimensionality estimator exploiting the norm angle and concentration. Pattern recognition. 472569-2581 (2014).
Hein, M. & Audibert, J.-Y. Estimating the intrinsic dimensionality of submanifolds in RD. In proc. 22nd International Conference on Machine Learning 289–296 (Association for Computing Machinery, 2005).
Grassberger, P. & Procaccia, I. en Chaotic attractor theory 170–189 (Springer, 2004).
Pukrittayakamee, A. et al. Simultaneous adjustment of a potential energy surface and its corresponding force fields using feedforward neural networks. J. Chem. Phys. 130134101 (2009).
Wu, J., Lim, JJ, Zhang, H., Tenenbaum, JB & Freeman, WT Physics 101: Learning Physical Properties of Objects from Unlabeled Videos. In proc. British Computer Vision Conference (BMVC) (eds Wilson, RC et al.) 39.1-39.12 (BMVA Press, 2016).
Chmiela, S. et al. Machine learning of precise, energy-efficient molecular force fields. Science. Adv. 3e1603015 (2017).
Schütt, KT, Arbabzadah, F., Chmiela, S., Müller, KR, and Tkatchenko, A. Insights into quantum chemistry from deep tensor neural networks. Nat. Common. 813890 (2017).
Smith, JS, Isayev, O. & Roitberg, AE ANI-1: A Scalable Neural Network Potential with DFT Accuracy at Force Field Computational Cost. Chem. Science. 83192–3203 (2017).
Lutter, M., Ritter, C. & Peters, J. Deep Lagrangian Networks: Using Physics as a Preliminary Model for Deep Learning. In International Conference on Representations of Learning (2019).
Bondesan, R. & Lamacraft, A. Learning the symmetries of classical integrable systems. Preprint at https://arxiv.org/abs/1906.04645 (2019).
Greydanus, SJ, Dzumba, M. & Yosinski, J. Hamiltonian Neural Networks. Preprint at https://arxiv.org/abs/1906.01563 (2019).
Swischuk, R., Kramer, B., Huang, C., and Willcox, K. Training physics-based reduced-order models for a single-injector combustion process. AIAA J. 582658–2672 (2020).
Lange, H., Brunton, SL & Kutz, JN From Fourier to Koopman: Spectral methods for the prediction of long-term time series. J.Mach. Learn. Res. 221–38 (2021).
Mallen, A., Lange, H. & Kutz, JN Koopman Deep probabilistic: long-term time series forecasting under periodic uncertainties. Preprint at https://arxiv.org/abs/2106.06033 (2021).
Chen B et al. Dataset for the article titled Discovering State Variables Hidden in Experimental Data (1.0). Zenodo https://doi.org/10.5281/zenodo.6653856 (2022).
Chen B et al. BoyuanChen/neural state variables: (v1.0). Zenodo https://doi.org/10.5281/zenodo.6629185 (2022).