Abarbanel, H. D. I., Rozdeba, P. J., and Shirman, S.: Machine Learning: Deepest
Learning as Statistical Data Assimilation Problems, Neural Comput., 30,
2025–2055, https://doi.org/10.1162/neco_a_01094, 2018. a, b

Amezcua, J., Goodliff, M., and van Leeuwen, P.-J.: A weak-constraint
4DEnsembleVar. Part I: formulation and simple model experiments, Tellus
A, 69, 1271564, https://doi.org/10.1080/16000870.2016.1271564, 2017. a

Asch, M., Bocquet, M., and Nodet, M.: Data Assimilation: Methods,
Algorithms, and Applications, Fundamentals of Algorithms, SIAM,
Philadelphia, 2016. a, b

Aster, R. C., Borchers, B., and Thuber, C. H.: Parameter Estimation and
Inverse Problems, Elsevier Academic Press, 2nd Edn., 2013. a

Bocquet, M.: Parameter field estimation for atmospheric dispersion:
Application to the Chernobyl accident using 4D-Var, Q. J. Roy.
Meteor. Soc., 138, 664–681, https://doi.org/10.1002/qj.961, 2012. a, b

Brunton, S. L., Proctor, J. L., and Kutz, J. N.: Discovering governing
equations from data by sparse identification of nonlinear dynamical systems,
P. Natl. Acad. Sci. USA, 113, 3932–3937, https://doi.org/10.1073/pnas.1517384113, 2016. a, b

Buizza, R., Miller, M., and Palmer, T. N.: Stochastic representation of model
uncertainties in the ECMWF ensemble prediction system, Q. J. Roy. Meteor.
Soc., 125, 2887–2908, 1999. a

Byrd, R. H., Lu, P., and Nocedal, J.: A Limited Memory Algorithm for Bound
Constrained Optimization, SIAM J. Sci. Stat.
Comp., 16, 1190–1208, 1995. a

Carlu, M., Ginelli, F., Lucarini, V., and Politi, A.: Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model, Nonlin. Processes Geophys., 26, 73–89, https://doi.org/10.5194/npg-26-73-2019, 2019. a

Carrassi, A. and Vannitsem, S.: Accounting for model error in variational data
assimilation: A deterministic formulation, Mon. Weather Rev., 138, 3369–3386,
https://doi.org/10.1175/2010MWR3192.1, 2010. a

Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G.: Data Assimilation in
the Geosciences: An overview on methods, issues, and perspectives, WIREs
Climate Change, 9, e535, https://doi.org/10.1002/wcc.535, 2018. a

Chang, B., Meng, L., Haber, E., Tung, F., and Begert, D.: Multi-level residual
networks from dynamical systems view, in: Proceedings of ICLR 2018, 2018. a

Chen, T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D.: Neural ordinary differential equations, in: Advances in Neural Information Processing Systems, 6571–6583, 2018. a

Dreano, D., Tandeo, P., Pulido, M., Ait-El-Fquih, B., Chonavel, T., and Hoteit,
I.: Estimating model error covariances in nonlinear state-space models using
Kalman smoothing and the expectation-maximisation algorithm, Q. J. Roy.
Meteor. Soc., 143, 1877–1885, https://doi.org/10.1002/qj.3048, 2017. a

Dueben, P. D. and Bauer, P.: Challenges and design choices for global weather and climate models based on machine learning, Geosci. Model Dev., 11, 3999–4009, https://doi.org/10.5194/gmd-11-3999-2018, 2018. a

Fablet, R., Ouala, S., and Herzet, C.: Bilinear residual neural network for the
identification and forecasting of dynamical systems, in: EUSIPCO 2018,
European Signal Processing Conference, Rome, Italy, 1–5,
available at: https://hal.archives-ouvertes.fr/hal-01686766 (last access: 8 July 2019), 2018. a, b, c, d

Gautschi, W.: Numerical analysis, Springer Science & Business Media, 2nd
Edn., 2012. a

Goodfellow, I., Bengio, Y., and Courville, A.: Deep learning, The MIT Press,
Cambridge Massachusetts, London, England, 2016. a

Grudzien, C., Carrassi, A., and Bocquet, M.: Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error, Nonlin. Processes Geophys., 25, 633–648, https://doi.org/10.5194/npg-25-633-2018, 2018. a

Harlim, J.: Model error in data assimilation, in: Nonlinear and stochastic
climate dynamics, edited by: Franzke, C. L. E. and O'Kane, T. J.,
Cambridge University Press, 276–317, https://doi.org/10.1017/9781316339251.011, 2017. a

Harlim, J.: Data-driven computational methods: parameter and operator
estimations, Cambridge University Press, Cambridge, 2018. a

Hodyss, D.: Ensemble State Estimation for Nonlinear Systems Using Polynomial
Expansions in the Innovation, Mon. Weather Rev., 139, 3571–3588,
https://doi.org/10.1175/2011MWR3558.1, 2011. a

Hodyss, D.: Accounting for Skewness in Ensemble Data Assimilation, Mon. Weather
Rev., 140, 2346–2358, https://doi.org/10.1175/MWR-D-11-00198.1, 2012. a

Hsieh, W. W. and Tang, B.: Applying Neural Network Models to Prediction and
Data Analysis in Meteorology and Oceanography, B. Am. Meteorol. Soc.,
79, 1855–1870, https://doi.org/10.1175/1520-0477(1998)079<1855:ANNMTP>2.0.CO;2, 1998. a

Janjić, T., Bormann, N., Bocquet, M., Carton, J. A., Cohn, S. E., Dance,
S. L., Losa, S. N., Nichols, N. K., Potthast, R., Waller, J. A., and Weston,
P.: On the representation error in data assimilation, Q. J. Roy. Meteor.
Soc., 144, 1257–1278, https://doi.org/10.1002/qj.3130, 2018. a

Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability,
Cambridge University Press, Cambridge, 2002. a

Kassam, A.-K. and Trefethen, L. N.: Fourth-Order Time-Stepping For Stiff
PDEs, Siam J. Sci. Comput., 26, 1214–1233,
https://doi.org/10.1137/S1064827502410633, 2005. a

Kondrashov, D. and Chrekroun, M. D.: Data-adaptive harmonic spectra and
multilayer Stuart-Landau models, Chaos, 27, 093110,
https://doi.org/10.1016/j.physd.2014.12.005, 2017. a

Kondrashov, D., Chrekroun, M. D., and Ghil, M.: Data-driven non-Markovian
closure models, Physica D, 297, 33–55,
https://doi.org/10.1063/1.4989400, 2015. a

Kondrashov, D., Chrekroun, M. D., Yuan, X., and Ghil, M.: Data-adaptive
harmonic decomposition and stochastic modeling of Arctic sea ice, in:
Advances in Nonlinear Geosciences, edited by: Tsonis, A.,
Springer, Cham, 179–205, https://doi.org/10.1007/978-3-319-58895-7_10, 2018. a

Kuramoto, Y. and Tsuzuki, T.: Persistent propagation of concentration waves in
dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55,
356–369, 1976. a

Lguensat, R., Tandeo, P., Ailliot, P., Pulido, M., and Fablet, R.: The Analog
Data Assimilation, Mon. Weather Rev., 145, 4093–4107,
https://doi.org/10.1175/MWR-D-16-0441.1, 2017. a

Long, Z., Lu, Y., Ma, X., and Dong, B.: PDE-Net: Learning PDEs from Data,
in: Proceedings of the 35th International Conference on Machine Learning,
2018. a

Lorenz, E. N.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141,
1963. a

Lorenz, E. N.: Designing Chaotic Models, J. Atmos. Sci., 62, 1574–1587,
https://doi.org/10.1175/JAS3430.1, 2005. a

Lorenz, E. N. and Emanuel, K. A.: Optimal sites for supplementary weather
observations: simulation with a small model, J. Atmos. Sci., 55, 399–414,
https://doi.org/10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2, 1998. a

Magnusson, L. and Källén, E.: Factors influencing skill improvements in
the ECMWF forecasting system, Mon. Weather Rev., 141, 3142–3153,
https://doi.org/10.1175/MWR-D-12-00318.1, 2013. a

Mitchell, L. and Carrassi, A.: Accounting for model error due to unresolved
scales within ensemble Kalman filtering, Q. J. Roy. Meteor. Soc., 141,
1417–1428, https://doi.org/10.1175/MWR-D-16-0478.1, 2015. a

Paduart, J., Lauwers, L., Swevers, J., Smolders, K., Schoukens, J., and
Pintelon, R.: Identification of nonlinear systems using polynomial nonlinear
state space models, Automatica, 46, 647–656,
https://doi.org/10.1016/j.automatica.2010.01.001, 2010. a, b

Park, D. C. and Zhu, Y.: Bilinear recurrent neural network, IEEE World Congress on Computational Intelligence, Neural
Networks, 3, 1459–1464, 1994. a

Pathak, J., Lu, Z., Hunt, B. R., Girvan, M., and Ott, E.: Using machine
learning to replicate chaotic attractors and calculate Lyapunov exponents
from data, Chaos, 27, 121102, https://doi.org/10.1063/1.5010300, 2017.
a

Pathak, J., Hunt, B., Girvan, M., Lu, Z., and Ott, E.: Model-Free Prediction of
Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing
Approach, Phys. Rev. Lett., 120, 024102,
https://doi.org/10.1103/PhysRevLett.120.024102, 2018. a

Pulido, M., Tandeo, P., Bocquet, M., Carrassi, A., and Lucini, M.: Stochastic
parameterization identification using ensemble Kalman filtering combined
with maximum likelihood methods, Tellus A, 70, 1442099,
https://doi.org/10.1080/16000870.2018.1442099, 2018. a, b, c, d, e

Raanes, P. N., Carrassi, A., and Bertino, L.: Extending the square root method
to account for additive forecast noise in ensemble methods, Mon. Weather Rev.,
143, 3857–3873, https://doi.org/10.1175/MWR-D-14-00375.1, 2015. a

Raanes, P. N., Bocquet, M., and Carrassi, A.: Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures, Q. J. Roy. Meteor.
Soc., 145, 53–75, https://doi.org/10.1002/qj.3386, 2019. a

Resseguier, V., Mémin, E., and Chapron, B.: Geophysical flows under
location uncertainty, Part I Random transport and general models,
Geophys. Astro. Fluid, 111, 149–176,
https://doi.org/10.1080/03091929.2017.1310210, 2017. a

Ruiz, J. J., Pulido, M., and Miyoshi, T.: Estimating model parameters with
ensemble-based data assimilation: A Review, J. Meteorol. Soc. Jpn., 91,
79–99, https://doi.org/10.2151/jmsj.2013-201, 2013. a

Sakov, P., Haussaire, J.-M., and Bocquet, M.: An iterative ensemble Kalman
filter in presence of additive model error, Q. J. Roy. Meteor. Soc., 144,
1297–1309, https://doi.org/10.1002/qj.3213, 2018. a

Sivashinsky, G. I.: Nonlinear analysis of hydrodynamic instability in laminar
flames-I. Derivation of basic equations, Acta Astronaut., 4, 1177–1206,
1977. a

Tandeo, P., Ailliot, P., Bocquet, M., Carrassi, A., Miyoshi, T., Pulido, M.,
and Zhen, Y.: Joint Estimation of Model and Observation Error Covariance
Matrices in Data Assimilation: a Review,
available at: https://hal-imt-atlantique.archives-ouvertes.fr/hal-01867958 (last access: 8 July 2019),
submitted, 2019. a

Trémolet, Y.: Accounting for an imperfect model in 4D-Var, Q. J. Roy.
Meteor. Soc., 132, 2483–2504, 2006. a

Wang, Y.-J. and Lin, C.-T.: Runge-Kutta neural network for identification
of dynamical systems in high accuracy, IEEE T. Neural Networ.,
9, 294–307, https://doi.org/10.1109/72.661124, 1998. a, b

Weinan, E.: A proposal on machine learning via dynamical systems, Commun. Math.
Stat., 5, 1–11, https://doi.org/10.1007/s40304-017-0103-z, 2017. a

Whitaker, J. S. and Hamill, T. M.: Evaluating Methods to Account for System
Errors in Ensemble Data Assimilation, Mon. Weather Rev., 140, 3078–3089,
https://doi.org/10.1175/MWR-D-11-00276.1, 2012. a