Preeti
Summarise this content to 300 words
Brezis, H. & Browder, F. Partial differential equations in the 20th century. Adv. Math. 135, 76–144 (1998).
Dissanayake, M. & Phan-Thien, N. Neural-network-based approximations for solving partial differential equations. Commun. Numer. Methods Eng. 10, 195–201 (1994).
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 1522–1525 (IEEE, 1993).
González-García, R., Rico-Martìnez, R. & Kevrekidis, I. G. Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965–S968 (1998).
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, 686–707 (2019).
Yu, B. et al. The Deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1–12 (2018).
Müller, 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).
Bruna, J., Peherstorfer, B. & Vanden-Eijnden, E. Neural Galerkin schemes with active learning for high-dimensional evolution equations. J. Comput. Phys. 496, 112588 (2024).
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 8459–8468 (PMLR, 2020).
Burger, M. et al. Connections between deep learning and partial differential equations. Eur. J. Appl. Math. 32, 395–396 (2021).
Loiseau, J.-C. & Brunton, S. L. Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67 (2018).
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, 477–508 (2020).
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 1457–1466 (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 2241–2256 (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, 422–440 (2021).
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, 9943–9948 (2007).
Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 324, 81–85 (2009).
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, 3932–3937 (2016).
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).
Schaeffer, H. Learning partial differential equations via data discovery and sparse optimization. Proc. Math. Phys. Eng. Sci. 473, 20160446 (2017).
Zanna, L. & Bolton, T. Data-driven equation discovery of ocean mesoscale closures. Geophys. Res. Lett. 47, e2020GL088376 (2020).
Schmelzer, M., Dwight, R. P. & Cinnella, P. Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow Turbulence Combustion 104, 579–603 (2020).
Beetham, S. & Capecelatro, J. Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids 5, 084611 (2020).
Beetham, S., Fox, R. O. & Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid-particle flows. J. Fluid Mech. 914, A11 (2021).
Bakarji, J. & Tartakovsky, D. M. Data-driven discovery of coarse-grained equations. J. Comput. Phys. 434, 110219 (2021).
Maslyaev, M., Hvatov, A. & Kalyuzhnaya, A. Data-driven partial derivative equations discovery with evolutionary approach. In Proc. Computational Science–ICCS 2019: 19th International Conference Part V 19, 635–641 (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).
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).
Xu, H., Zhang, D. & Zeng, J. Deep-learning of parametric partial differential equations from sparse and noisy data. Phys. Fluids 33, 037132 (2021).
Xu, H. & Zhang, D. Robust discovery of partial differential equations in complex situations. Phys. Rev. Res. 3, 033270 (2021).
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).
Taira, K. & Colonius, T. The immersed boundary method: a projection approach. J. Comput. Phys. 225, 2118–2137 (2007).
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, 2131–2146 (2008).
Van Breugel, F., Kutz, J. N. & Brunton, B. W. Numerical differentiation of noisy data: a unifying multi-objective optimization framework. IEEE Access 8, 196865–196877 (2020).
Messenger, D. A. & Bortz, D. M. Weak SINDy: Galerkin-based data-driven model selection. Multiscale Model. Simul. 19, 1474–1497 (2021).
Messenger, D. A. & Bortz, D. M. Weak SINDy for partial differential equations. J. Comput. Phys. 443, 110525 (2021).
Schaeffer, H. & McCalla, S. G. Sparse model selection via integral terms. Phys. Rev. E 96, 023302 (2017).
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).
Gurevich, D. R., Reinbold, P. A. & Grigoriev, R. O. Robust and optimal sparse regression for nonlinear PDE models. Chaos 29, 103113 (2019).
Alves, E. P. & Fiuza, F. Data-driven discovery of reduced plasma physics models from fully kinetic simulations. Phys. Rev. Res. 4, 033192 (2022).
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).
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).
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).
Pope, S. A more general effective-viscosity hypothesis. J. Fluid Mech. 72, 331–340 (1975).
Ling, J., Kurzawski, A. & Templeton, J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016).
Duraisamy, K., Iaccarino, G. & Xiao, H. Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377 (2019).
Ahmed, S. E. et al. On closures for reduced order models—a spectrum of first-principle to machine-learned avenues. Phys. Fluids 33, 091301 (2021).
Supekar, R. et al. Learning hydrodynamic equations for active matter from particle simulations and experiments. Proc. Natl Acad. Sci. USA 120, e2206994120 (2023).
Kaptanoglu, A. A. et al. PySINDy: a comprehensive Python package for robust sparse system identification. J. Open Source Softw. 7, 3994 (2022).
Long, Z., Lu, Y., Ma, X. & Dong, B. PDE-Net: learning PDEs from data. In Proc. International Conference on Machine Learning 3208–3216 (PMLR, 2018).
Long, Z., Lu, Y. & Dong, B. PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network. J. Comput. Phys. 399, 108925 (2019).
Atkinson, S. Bayesian hidden physics models: uncertainty quantification for discovery of nonlinear partial differential operators from data. Preprint at https://arxiv.org/abs/2006.04228 (2020).
Cai, J.-F., Dong, B., Osher, S. & Shen, Z. Image restoration: total variation, wavelet frames and beyond. J. Am. Math. Soc. 25, 1033–1089 (2012).
Dong, B., Jiang, Q. & Shen, Z. Image restoration: wavelet frame shrinkage, nonlinear evolution PDEs and beyond. Multiscale Model. Simul. 15, 606–660 (2017).
Schaeffer, H., Caflisch, R., Hauck, C. D. & Osher, S. Sparse dynamics for partial differential equations. Proc. Natl Acad. Sci. USA 110, 6634–6639 (2013).
Cranmer, M. D., Xu, R., Battaglia, P. & Ho, S. Learning symbolic physics with graph networks. Preprint at https://arxiv.org/abs/1909.05862 (2019).
Cranmer, M. et al. Discovering symbolic models from deep learning with inductive biases. In Advances in Neural Information Processing Systems (eds Larochelle, H. et al.) 17429–17442 (Curran Associates, Inc., 2020).
Callaham, J. L., Koch, J. V., Brunton, B. W., Kutz, J. N. & Brunton, S. L. Learning dominant physical processes with data-driven balance models. Nat. Commun. 12, 1016 (2021).
Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).
Bakarji, J., Callaham, J., Brunton, S. L. & Kutz, J. N. Dimensionally consistent learning with Buckingham Pi. Nat. Comput. Sci. 2, 834–844 (2022).
Koopman, B. O. Hamiltonian systems and transformation in Hilbert space. Proc. Natl Acad. Sci. USA 17, 315–318 (1931).
Mezić, I. & Banaszuk, A. Comparison of systems with complex behavior. Phys. D Nonlinear Phenomena 197, 101–133 (2004).
Mezić, I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325 (2005).
Mezić, I. Analysis of fluid flows via spectral properties of the Koopman operator. Ann. Rev. Fluid Mech. 45, 357–378 (2013).
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115–127 (2009).
Kutz, J. N., Brunton, S. L., Brunton, B. W. & Proctor, J. L. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems (SIAM, 2016).
Budišić, M., Mohr, R. & Mezić, I. Applied Koomanism. Chaos 22, 047510 (2012).
Brunton, S. L., Budišić, M., Kaiser, E. & Kutz, J. N. Modern Koopman theory for dynamical systems. SIAM Rev. 64, 229–340 (2022).
Cover, T. M. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Trans. Electron Comput. EC-14, 326–334 (1965).
Hopf, E. The partial differential equation ut + uux = μuxx. Commun. Pure Appl. Math. 3, 201–230 (1950).
Cole, J. D. On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951).
Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. The inverse scattering transform-fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974).
Ablowitz, M. J. & Segur, H. Solitons and the Inverse Scattering Transform (SIAM, 1981).
Lusch, B., Kutz, J. N. & Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9, 4950 (2018).
Wehmeyer, C. & Noé, F. Time-lagged autoencoders: deep learning of slow collective variables for molecular kinetics. J. Chem. Phys 148, 241703 (2018).
Mardt, A., Pasquali, L., Wu, H. & Noé, F. VAMPnets: deep learning of molecular kinetics. Nat. Commun. 9, 5 (2018).
Takeishi, N., Kawahara, Y. & Yairi, T. Learning Koopman invariant subspaces for dynamic mode decomposition. In Advances in Neural Information Processing Systems 1130–1140 (NIPS, 2017).
Yeung, E., Kundu, S. & Hodas, N. Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. Preprint at https://arxiv.org/abs/1708.06850 (2017).
Otto, S. E. & Rowley, C. W. Linearly recurrent autoencoder networks for learning dynamics. SIAM J. Appl. Dyn. Syst. 18, 558–593 (2019).
Li, Q., Dietrich, F., Bollt, E. M. & Kevrekidis, I. G. Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator. Chaos 27, 103111 (2017).
Eivazi, H., Guastoni, L., Schlatter, P., Azizpour, H. & Vinuesa, R. Recurrent neural networks and Koopman-based frameworks for temporal predictions in a low-order model of turbulence. Int. J. Heat Fluid Flow 90, 108816 (2021).
Gin, C., Lusch, B., Brunton, S. L. & Kutz, J. N. Deep learning models for global coordinate transformations that linearise PDEs. Eur. J. Appl. Math. 32, 515–539 (2021).
Noé, F. & Nuske, F. A variational approach to modeling slow processes in stochastic dynamical systems. Multiscale Model. Simul. 11, 635–655 (2013).
Nüske, F., Keller, B. G., Pérez-Hernández, G., Mey, A. S. & Noé, F. Variational approach to molecular kinetics. J.Chem. Theory Comput. 10, 1739–1752 (2014).
Williams, M. O., Kevrekidis, I. G. & Rowley, C. W. A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 6, 1307–1346 (2015).
Williams, M. O., Rowley, C. W. & Kevrekidis, I. G. A kernel approach to data-driven Koopman spectral analysis. J. Comput. Dyn. 2, 247–265 (2015).
Klus, S. Data-driven model reduction and transfer operator approximation. J. Nonlinear Sci. 28, 985–1010 (2018).
Kutz, J. N., Proctor, J. L. & Brunton, S. L. Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems. Complexity 2018, 6010634 (2018).
Page, J. & Kerswell, R. R. Koopman analysis of burgers equation. Phys. Rev. Fluids 3, 071901 (2018).
Lu, L., Jin, P. & Karniadakis, G. E. DeepONet: learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. Preprint at https://arxiv.org/abs/1910.03193 (2019).
Lu, L., Jin, P., Pang, G., Zhang, Z. & Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3, 218–229 (2021).
Kovachki, N. et al. Neural operator: learning maps between function spaces with applications to PDEs. J. Mach. Learn. Res. 24, 1–97 (2023).
He, K., Zhang, X., Ren, S. & Sun, J. Deep residual learning for image recognition. In Proc. IEEE Conference on Computer Vision and Pattern Recognition 770–778 (IEEE, 2016).
Colbrook, M. J., Ayton, L. J. & Szőke, M. Residual dynamic mode decomposition: robust and verified Koopmanism. J. Fluid Mech. 955, A21 (2023).
Colbrook, M. J. & Townsend, A. Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. Commun. Pure Appl. Math. 77, 221–283 (2024).
Lumley, J. Toward a turbulent constitutive relation. J. Fluid Mech. 41, 413–434 (1970).
Berkooz, G., Holmes, P. & Lumley, J. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575 (1993).
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. Turbulence, Coherent Structures, Dynamical Systems and Symmetry 2nd edn (Cambridge Univ. Press, 2012).
Taira, K. et al. Modal analysis of fluid flows: an overview. AIAA J. 55, 4013–4041 (2017).
Taira, K. et al. Modal analysis of fluid flows: applications and outlook. AIAA J. 58, 998–1022 (2020).
Benner, P., Gugercin, S. & Willcox, K. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57, 483–531 (2015).
Qian, E., Kramer, B., Peherstorfer, B. & Willcox, K. Lift & Learn: physics-informed machine learning for large-scale nonlinear dynamical systems. Phys. D Nonlinear Phenom. 406, 132401 (2020).
Peherstorfer, B. & Willcox, K. Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016).
Benner, P., Goyal, P., Kramer, B., Peherstorfer, B. & Willcox, K. Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Comput. Methods Appl. Mech. Eng. 372, 113433 (2020).
Peherstorfer, B., Drmac, Z. & Gugercin, S. Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points. SIAM J. Sci. Comput. 42, A2837–A2864 (2020).
Holmes, P. & Guckenheimer, J. (eds) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Vol. 42 (Springer, 1983).
Rowley, C. W. & Marsden, J. E. Reconstruction equations and the Karhunen-Loève expansion for systems with symmetry. Phys. D Nonlinear Phenom. 142, 1–19 (2000).
Abraham, R., Marsden, J. E. & Ratiu, T. Manifolds, Tensor Analysis and Applications Vol. 75 (Springer, 1988).
Marsden, J. E. & Ratiu, T. S. Introduction to Mechanics and Symmetry 2nd edn (Springer, 1999).
Lee, K. & Carlberg, K. T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404, 108973 (2020).
Schmid, P. J. Dynamic mode decomposition for numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010).
Loiseau, J.-C., Noack, B. R. & Brunton, S. L. Sparse reduced-order modeling: sensor-based dynamics to full-state estimation. J. Fluid Mech. 844, 459–490 (2018).
Deng, N., Noack, B. R., Morzynski, M. & Pastur, L. R. Low-order model for successive bifurcations of the fluidic pinball. J. Fluid Mech. 884, A37 (2020).
Loiseau, J.-C. Data-driven modeling of the chaotic thermal convection in an annular thermosyphon.Theor. Comput. Fluid Dyn 34, 339–365 (2020).
Guan, Y., Brunton, S. L. & Novosselov, I. Sparse nonlinear models of chaotic electroconvection. R. Soc. Open Sci. 8, 202367 (2021).
Kaptanoglu, A. A., Morgan, K. D., Hansen, C. J. & Brunton, S. L. Physics-constrained, low-dimensional models for magnetohydrodynamics: first-principles and data-driven approaches. Phys. Rev. E 104, 015206 (2021).
Kaptanoglu, A. A., Callaham, J. L., Aravkin, A., Hansen, C. J. & Brunton, S. L. Promoting global stability in data-driven models of quadratic nonlinear dynamics. Phys. Rev. Fluids 6, 094401 (2021).
Deng, N., Noack, B. R., Morzyński, M. & Pastur, L. R. Galerkin force model for transient and post-transient dynamics of the fluidic pinball. J. Fluid Mech. 918, A4 (2021).
Callaham, J. L., Loiseau, J.-C., Rigas, G. & Brunton, S. L. Nonlinear stochastic modelling with Langevin regression. Proc. R. Soc. A 477, 20210092 (2021).
Callaham, J. L., Brunton, S. L. & Loiseau, J.-C. On the role of nonlinear correlations in reduced-order modeling. J. Fluid Mech. 938, A1 (2022).
Callaham, J. L., Rigas, G., Loiseau, J.-C. & Brunton, S. L. An empirical mean-field model of symmetry-breaking in a turbulent wake. Sci. Adv. 8, eabm4786 (2022).
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).
Pathak, J., Hunt, B., Girvan, M., Lu, Z. & Ott, E. Model-free prediction of large spatiotemporally chaotic systems from data: a reservoir computing approach. Phys. Rev. Lett. 120, 024102 (2018).
Champion, K., Lusch, B., Kutz, J. N. & Brunton, S. L. Data-driven discovery of coordinates and governing equations. Proc. Natl Acad. Sci. USA 116, 22445–22451 (2019).
Baddoo, P. J., Herrmann, B., McKeon, B. J. & Brunton, S. L. Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization (LANDO). Proc. R. Soc. A 478, 20210830 (2022).
Reed, M. & Simon, B. Methods of Modern Mathematical Physics. I 2nd edn (Academic, 1980).
Courant, R. & Hilbert, D. Methods of Mathematical Physics: Partial Differential Equations (Wiley, 2008).
Li, Z. et al. Neural operator: graph kernel network for partial differential equations. Preprint at https://arxiv.org/abs/2003.03485 (2020).
Li, Z. et al. Multipole graph neural operator for parametric partial differential equations. Preprint at https://arxiv.org/abs/2006.09535 (2020).
Li, Z. et al. Fourier neural operator for parametric partial differential equations. Preprint at https://arxiv.org/abs/2010.08895 (2020).
Gin, C. R., Shea, D. E., Brunton, S. L. & Kutz, J. N. DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems. Sci. Rep. 11, 21614 (2021).
de Hoop, M. V., Kovachki, N. B., Nelsen, N. H. & Stuart, A. M. Convergence rates for learning linear operators from noisy data. SIAM-ASA J. Uncertain. Quantif. 11, 480–513 (2023).
De Hoop, M., Huang, D. Z., Qian, E. & Stuart, A. M. The cost-accuracy trade-off in operator learning with neural networks. Preprint at https://arxiv.org/abs/2203.13181 (2022).
Mollenhauer, M., Mücke, N. & Sullivan, T. Learning linear operators: infinite-dimensional regression as a well-behaved non-compact inverse problem. Preprint at https://arxiv.org/abs/2211.08875 (2022).
Lange, H., Brunton, S. L. & Kutz, J. N. From Fourier to Koopman: spectral methods for long-term time series prediction. J. Mach. Learn. Res. 22, 1–38 (2021).
Lanthaler, S., Mishra, S. & Karniadakis, G. E. Error estimates for DeepONets: a deep learning framework in infinite dimensions. Trans. Math. Appl. 6, tnac001 (2022).
Kovachki, N., Lanthaler, S. & Mishra, S. On universal approximation and error bounds for Fourier neural operators. J. Mach. Learn. Res. 22, 13237–13312 (2021).
Lyu, Y., Zhao, X., Gong, Z., Kang, X. & Yao, W. Multi-fidelity prediction of fluid flow based on transfer learning using Fourier neural operator. Phys. Fluids 35, 077118 (2023).
Gopakumar, V. et al. Plasma surrogate modelling using Fourier neural operators. Nucl. Fusion 64, 056025 (2024).
Kurth, T. et al. FourCastNet: accelerating global high-resolution weather forecasting using adaptive Fourier neural operators. In Proc. Platform for Advanced Scientific Computing Conference 1–11 (ACM, 2023).
Raonić, B. et al. Convolutional neural operators for robust and accurate learning of PDEs. In Advances in Neural Information Processing Systems (eds Oh, A. et al.) 77187–77200 (NeurIPS, 2023).
Fanaskov, V. & Oseledets, I. Spectral neural operators. Preprint at https://arxiv.org/abs/2205.10573 (2022).
Chen, T. & Chen, H. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans. Neural Networks 6, 911–917 (1995).
Cao, S. Choose a transformer: Fourier or Galerkin. Adv. Neural Inf. Process. Syst. 34, 24924–24940 (2021).
Li, Z., Meidani, K. & Farimani, A. B. Transformer for partial differential equations’ operator learning. Transact. Mach. Learn. Res. https://openreview.net/pdf?id=EPPqt3uERT (2023).
Vinuesa, R. & Brunton, S. L. Enhancing computational fluid dynamics with machine learning. Nat. Comput. Sci. 2, 358–366 (2022).
Mishra, S. A machine learning framework for data driven acceleration of computations of differential equations. Math. Eng. 1, 118–146 (2019).
Bar-Sinai, Y., Hoyer, S., Hickey, J. & Brenner, M. P. Learning data-driven discretizations for partial differential equations. Proc. Natl Acad. Sci. USA 116, 15344–15349 (2019).
Stevens, B. & Colonius, T. Enhancement of shock-capturing methods via machine learning. Theor. Comput. Fluid Dyn. 34, 483–496 (2020).
Barrault, M., Maday, Y., Nguyen, N. C. & Patera, A. T. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339, 667–672 (2004).
Chaturantabut, S. & Sorensen, D. C. Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010).
Stevens, B. & Colonius, T. FiniteNet: a fully convolutional LSTM network architecture for time-dependent partial differential equations. Preprint at https://arxiv.org/abs/2002.03014 (2020).
Yousif, M. Z., Zhang, M., Yu, L., Vinuesa, R. & Lim, H. A transformer-based synthetic-inflow generator for spatially developing turbulent boundary layers. J. Fluid Mech. 957, A6 (2023).
Kochkov, D. et al. Machine learning-accelerated computational fluid dynamics. Proc. Natl Acad. Sci. USA 118, e2101784118 (2021).
Fukami, K., Fukagata, K. & Taira, K. Super-resolution reconstruction of turbulent flows with machine learning. J. Fluid Mech. 870, 106–120 (2019).
Fukami, K., Fukagata, K. & Taira, K. Machine-learning-based spatio-temporal super resolution reconstruction of turbulent flows. J. Fluid Mech. 909, A9 (2021).
Fukami, K. & Taira, K. Robust machine learning of turbulence through generalized Buckingham Pi-inspired pre-processing of training data. In APS Division of Fluid Dynamics Meeting Abstracts A31-004 (APS, 2021).
Pan, S., Brunton, S. L. & Kutz, J. N. Neural implicit flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data. J. Mach. Learn. Res. 24, 1607–1666 (2023).
Takamoto, M. et al. PDEBench: an extensive benchmark for scientific machine learning. In Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (NeurIPS, 2022).
Source link
Source link: https://www.nature.com/articles/s43588-024-00643-2
GIPHY App Key not set. Please check settings