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Brezis, H. & Browder, F. Partial differential equations in the 20th century. Adv. Math. 135, 76144 (1998).

Article MathSciNet Google Scholar

Dissanayake, M. & Phan-Thien, N. Neural-network-based approximations for solving partial differential equations. Commun. Numer. Methods Eng. 10, 195201 (1994).

Article Google Scholar

Rico-Martinez, R. & Kevrekidis, I. G. Continuous time modeling of nonlinear systems: a neural network-based approach. In Proc. IEEE International Conference on Neural Networks 15221525 (IEEE, 1993).

Gonzlez-Garca, R., Rico-Martnez, R. & Kevrekidis, I. G. Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965S968 (1998).

Article Google Scholar

Raissi, M., Perdikaris, P. & Karniadakis, G. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686707 (2019).

Article MathSciNet Google Scholar

Yu, B. et al. The Deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 112 (2018).

Article MathSciNet Google Scholar

Mller, J. & Zeinhofer, M. Deep Ritz revisited. Preprint at https://arxiv.org/abs/1912.03937 (2019).

Gao, H., Zahr, M. J. & Wang, J.-X. Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput. Methods Appl. Mech. Eng. 390, 114502 (2022).

Article MathSciNet Google Scholar

Bruna, J., Peherstorfer, B. & Vanden-Eijnden, E. Neural Galerkin schemes with active learning for high-dimensional evolution equations. J. Comput. Phys. 496, 112588 (2024).

Article MathSciNet Google Scholar

Battaglia, P. W. et al. Relational inductive biases, deep learning and graph networks. Preprint at https://arxiv.org/abs/1806.01261 (2018).

Sanchez-Gonzalez, A. et al. Learning to simulate complex physics with graph networks. In Proc. International Conference on Machine Learning 84598468 (PMLR, 2020).

Burger, M. et al. Connections between deep learning and partial differential equations. Eur. J. Appl. Math. 32, 395396 (2021).

Article Google Scholar

Loiseau, J.-C. & Brunton, S. L. Constrained sparse Galerkin regression. J. Fluid Mech. 838, 4267 (2018).

Article MathSciNet Google Scholar

Cranmer, M. et al. Lagrangian neural networks. Preprint at https://arxiv.org/abs/2003.04630 (2020).

Brunton, S. L., Noack, B. R. & Koumoutsakos, P. Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477508 (2020).

Article MathSciNet Google Scholar

Wang, R., Walters, R. & Yu, R. Incorporating symmetry into deep dynamics models for improved generalization. In International Conference on Learning Representations (ICLR, 2021).

Wang, R., Kashinath, K., Mustafa, M., Albert, A. & Yu, R. Towards physics-informed deep learning for turbulent flow prediction. In Proc. 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining 14571466 (ACM, 2020).

Brandstetter, J., Berg, R. V. D., Welling, M. & Gupta, J. K. Clifford neural layers for PDE modeling. In Eleventh International Conference on Learning Representations (ICLR, 2023)

De Haan, P., Weiler, M., Cohen, T. & Welling, M. Gauge equivariant mesh CNNS: anisotropic convolutions on geometric graphs. In International Conference on Learning Representations (ICLR, 2021).

Brandstetter, J., Welling, M. & Worrall, D. E. Lie point symmetry data augmentation for neural PDE solvers. In Proc. International Conference on Machine Learning 22412256 (PMLR, 2022).

Brandstetter, J., Worrall, D. & Welling, M. Message passing neural PDE solvers. Preprint at https://arxiv.org/abs/2202.03376 (2022).

Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422440 (2021).

Article Google Scholar

Brunton, S. L. & Kutz, J. N. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems and Control 2nd edn (Cambridge Univ. Press, 2022).

Bongard, J. & Lipson, H. Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104, 99439948 (2007).

Article Google Scholar

Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 324, 8185 (2009).

Article Google Scholar

Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 39323937 (2016).

Article MathSciNet Google Scholar

Cranmer, M. Interpretable machine learning for science with PySR and SymbolicRegression.jl. Preprint at https://arxiv.org/abs/2305.01582 (2023).

Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. Data-driven discovery of partial differential equations. Sci. Adv 3, e1602614 (2017).

Article Google Scholar

Schaeffer, H. Learning partial differential equations via data discovery and sparse optimization. Proc. Math. Phys. Eng. Sci. 473, 20160446 (2017).

MathSciNet Google Scholar

Zanna, L. & Bolton, T. Data-driven equation discovery of ocean mesoscale closures. Geophys. Res. Lett. 47, e2020GL088376 (2020).

Article Google Scholar

Schmelzer, M., Dwight, R. P. & Cinnella, P. Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow Turbulence Combustion 104, 579603 (2020).

Article Google Scholar

Beetham, S. & Capecelatro, J. Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids 5, 084611 (2020).

Article Google Scholar

Beetham, S., Fox, R. O. & Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid-particle flows. J. Fluid Mech. 914, A11 (2021).

Article MathSciNet Google Scholar

Bakarji, J. & Tartakovsky, D. M. Data-driven discovery of coarse-grained equations. J. Comput. Phys. 434, 110219 (2021).

Article MathSciNet Google Scholar

Maslyaev, M., Hvatov, A. & Kalyuzhnaya, A. Data-driven partial derivative equations discovery with evolutionary approach. In Proc. Computational ScienceICCS 2019: 19th International Conference Part V 19, 635641 (Springer, 2019).

Xu, H., Zhang, D. & Wang, N. Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data. J. Comput. Phys. 445, 110592 (2021).

Article MathSciNet Google Scholar

Xu, H., Chang, H. & Zhang, D. DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm. J. Comput. Phys. 418, 109584 (2020).

Article MathSciNet Google Scholar

Xu, H., Zhang, D. & Zeng, J. Deep-learning of parametric partial differential equations from sparse and noisy data. Phys. Fluids 33, 037132 (2021).

Article Google Scholar

Xu, H. & Zhang, D. Robust discovery of partial differential equations in complex situations. Phys. Rev. Res. 3, 033270 (2021).

Article Google Scholar

Chen, Y., Luo, Y., Liu, Q., Xu, H. & Zhang, D. Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE). Phys. Rev. Res. 4, 023174 (2022).

Article Google Scholar

Taira, K. & Colonius, T. The immersed boundary method: a projection approach. J. Comput. Phys. 225, 21182137 (2007).

Article MathSciNet Google Scholar

Colonius, T. & Taira, K. A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Methods Appl. Mech. Eng. 197, 21312146 (2008).

Article MathSciNet Google Scholar

Van Breugel, F., Kutz, J. N. & Brunton, B. W. Numerical differentiation of noisy data: a unifying multi-objective optimization framework. IEEE Access 8, 196865196877 (2020).

Article Google Scholar

Messenger, D. A. & Bortz, D. M. Weak SINDy: Galerkin-based data-driven model selection. Multiscale Model. Simul. 19, 14741497 (2021).

Article MathSciNet Google Scholar

Messenger, D. A. & Bortz, D. M. Weak SINDy for partial differential equations. J. Comput. Phys. 443, 110525 (2021).

Article MathSciNet Google Scholar

Schaeffer, H. & McCalla, S. G. Sparse model selection via integral terms. Phys. Rev. E 96, 023302 (2017).

Article MathSciNet Google Scholar

Fasel, U., Kutz, J. N., Brunton, B. W. & Brunton, S. L. Ensemble-SINDy: robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proc. R. Soc. A 478, 20210904 (2022).

Article MathSciNet Google Scholar

Gurevich, D. R., Reinbold, P. A. & Grigoriev, R. O. Robust and optimal sparse regression for nonlinear PDE models. Chaos 29, 103113 (2019).

Article MathSciNet Google Scholar

Alves, E. P. & Fiuza, F. Data-driven discovery of reduced plasma physics models from fully kinetic simulations. Phys. Rev. Res. 4, 033192 (2022).

Article Google Scholar

Reinbold, P. A., Gurevich, D. R. & Grigoriev, R. O. Using noisy or incomplete data to discover models of spatiotemporal dynamics. Phys. Rev. E 101, 010203 (2020).

Article Google Scholar

Suri, B., Kageorge, L., Grigoriev, R. O. & Schatz, M. F. Capturing turbulent dynamics and statistics in experiments with unstable periodic orbits. Phys. Rev. Lett. 125, 064501 (2020).

Article Google Scholar

Reinbold, P. A., Kageorge, L. M., Schatz, M. F. & Grigoriev, R. O. Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression. Nat. Commun. 12, 3219 (2021).

Article Google Scholar

Pope, S. A more general effective-viscosity hypothesis. J. Fluid Mech. 72, 331340 (1975).

Article Google Scholar

Ling, J., Kurzawski, A. & Templeton, J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155166 (2016).

Article MathSciNet Google Scholar

Duraisamy, K., Iaccarino, G. & Xiao, H. Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357377 (2019).

Article MathSciNet Google Scholar

Ahmed, S. E. et al. On closures for reduced order modelsa spectrum of first-principle to machine-learned avenues. Phys. Fluids 33, 091301 (2021).

Article Google Scholar

Supekar, R. et al. Learning hydrodynamic equations for active matter from particle simulations and experiments. Proc. Natl Acad. Sci. USA 120, e2206994120 (2023).

Article MathSciNet Google Scholar

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