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We propose derivative-informed neural operators (DINOs), a general family of neural networks to approximate operators as infinite-dimensional mappings from input function spaces to output function spaces or quantities of interest.
1909
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O. Bashir, K. Willcox, O. Ghattas, B. van Bloemen Waanders, and J. Hill, “Hessian-based model reduction for large-scale systems with initial condition inputs,” International Journal for Numerical Methods in Engineering , vol. 73, pp. 844–868, 2008
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N. H. Gokhale, P. E. Barbone, and A. A. Oberai, “Solution of the nonlinear elasticity imaging inverse problem: The compressible case,” Inverse Problems , vol. 24, no. 4, p. 045010, 2008
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O. Gonzalez and A. M. Stuart, A first course in continuum mechanics , ser. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2008. [Online]. Available: https://doi.org/10.1017/CBO9780511619571
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P. H. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders, and O. Ghattas, “Fast algorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations,” SIAM Journal on Scientific Computing , vol. 33, no. 1, pp. 407–432, 2011
2011
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N. Halko, P. G. Martinsson, and J. A. Tropp, “Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions,” SIAM Review , vol. 53, no. 2, pp. 217–288, 2011
2011
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F. Lindgren, H. Rue, and J. Lindström, “An explicit link between gaussian fields and gaussian markov random fields: the stochastic partial differential equation approach,” Journal of the Royal Statistical Society: Series B (Statistical Methodology) , vol. 73, no. 4, pp. 423–498, 2011
2011
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S. Goenezen, P. Barbone, and A. A. Oberai, “Solution of the nonlinear elasticity imaging inverse problem: The incompressible case,” Computer Methods in Applied Mechanics and Engineering , vol. 200, no. 13, pp. 1406–1420, 2011
2011
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J. Martin, L. C. Wilcox, C. Burstedde, and O. Ghattas, “A stochastic Newton mcmc method for large-scale statistical inverse problems with application to seismic inversion,” SIAM Journal on Scientific Computing , vol. 34, no. 3, pp. A1460–A1487, 2012
2012
Earlier work this paper cites.
T. Bui-Thanh, C. Burstedde, O. Ghattas, J. Martin, G. Stadler, and L. C. Wilcox, “Extreme-scale UQ for Bayesian inverse problems governed by PDEs,” in SC12: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis , 2012
2012
Earlier work this paper cites.
T. Bui-Thanh and O. Ghattas, “Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves,” Inverse Problems , vol. 28, no. 5, p. 055001, 2012
2012
Earlier work this paper cites.
——, “Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves,” Inverse Problems , vol. 28, no. 5, p. 055002, 2012
2012
Earlier work this paper cites.
J. Martin, L. C. Wilcox, C. Burstedde, and O. Ghattas, “A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion,” SIAM Journal on Scientific Computing , vol. 34, no. 3, pp. A1460–A1487, 2012
2012
Earlier work this paper cites.
——, “Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves,” Inverse Problems and Imaging , vol. 7, no. 4, pp. 1139–1155, 2013
2013
Earlier work this paper cites.
T. Bui-Thanh, O. Ghattas, J. Martin, and G. Stadler, “A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion,” SIAM Journal on Scientific Computing , vol. 35, no. 6, pp. A2494–A2523, 2013
2013
Earlier work this paper cites.
T. Bui-Thanh, O. Ghattas, J. Martin, and G. Stadler, “A computational framework for infinite-dimensional bayesian inverse problems part i: The linearized case, with application to global seismic inversion,” SIAM Journal on Scientific Computing , vol. 35, no. 6, pp. A2494–A2523, 2013
2013
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P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, “Automated derivation of the adjoint of high-level transient finite element programs,” SIAM Journal on Scientific Computing , vol. 35, no. 4, pp. C369–C393, 2013
2013
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N. Petra, J. Martin, G. Stadler, and O. Ghattas, “A computational framework for infinite-dimensional Bayesian inverse problems, part ii: Stochastic Newton MCMC with application to ice sheet flow inverse problems,” SIAM Journal on Scientific Computing , vol. 36, no. 4, pp. A1525–A1555, 2014
2014
Earlier work this paper cites.
A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, “A-optimal design of experiments for infinite-dimensional Bayesian linear inverse problems with regularized ℓ 0 \ell_{0} -sparsification,” SIAM Journal on Scientific Computing , vol. 36, no. 5, pp. A2122–A2148, 2014
2014
Earlier work this paper cites.
T. Bui-Thanh and O. Ghattas, “An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation,” SIAM/ASA Journal of Uncertainty Quantification , vol. 2, no. 1, pp. 203–222, 2014
2014
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P. G. Constantine, E. Dow, and Q. Wang, “Active subspace methods in theory and practice: applications to kriging surfaces,” SIAM Journal on Scientific Computing , vol. 36, no. 4, pp. A1500–A1524, 2014
2014
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D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” ICLR 2015 , 2014
2014
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——, “A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors,” Inverse Problems and Imaging , vol. 9, no. 1, pp. 27–54, 2015
2015
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T. Isaac, N. Petra, G. Stadler, and O. Ghattas, “Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet,” Journal of Computational Physics , vol. 296, pp. 348–368, September 2015
2015
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A. Baydin, B. Pearlmutter, A. Radul, and J. Siskind, “Automatic differentiation in machine learning: A survey. arXiv preprint arXiv: 150205767,” 2015
2015
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J.-S. Affagard, P. Feissel, and S. F. Bensamoun, “Identification of hyperelastic properties of passive thigh muscle under compression with an inverse method from a displacement field measurement,” Journal of Biomechanics , vol. 48, no. 15, pp. 4081–4086, 2015
2015
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M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, “The fenics project version 1.5,” Archive of Numerical Software , vol. 3, no. 100, 2015
2015
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S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini, and H. Zhang, “PETSc users manual,” Argonne National Laboratory, Tech. Rep. ANL-95/11 - Revision 3.6, 2015. [Online]. Available: http://www.mcs.anl.gov/petsc
P.-G. Martinsson and J. A. Tropp, “Randomized numerical linear algebra: Foundations and algorithms,” Acta Numerica , vol. 29, pp. 403–572, 2020
2020
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K. Bhattacharya, B. Hosseini, N. B. Kovachki, and A. M. Stuart, “Model reduction and neural networks for parametric PDEs,” SMAI Journal of Computational Mathematics, Volume 7 , 2021
2021
Later among the works it cites.
Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, “Fourier neural operator for parametric partial differential equations,” International Conference on Learning Representations , 2021
2021
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L. Lu, P. Jin, G. Pang, and G. E. Karniadakis, “DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators,” Nature Machine Intelligence , 2021
2021
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2015
Cited alongside, same era.
——, “A fast and scalable method for A-optimal design of experiments for infinite-dimensional Bayesian nonlinear inverse problems,” SIAM Journal on Scientific Computing , vol. 38, no. 1, pp. A243–A272, 2016
2016
Cited alongside, same era.
T. Cui, K. Law, and Y. Marzouk, “Dimension-independent likelihood-informed MCMC,” Journal of Computational Physics , vol. 304, pp. 109–137, 2016
2016
Cited alongside, same era.
2016
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A. Beskos, M. Girolami, S. Lan, P. E. Farrell, and A. M. Stuart, “Geometric MCMC for infinite-dimensional inverse problems,” Journal of Computational Physics , vol. 335, pp. 327–351, 2017
2017
Cited alongside, same era.
A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, “Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations,” SIAM/ASA Journal on Uncertainty Quantification , vol. 5, no. 1, pp. 1166–1192, 2017
2017
Cited alongside, same era.
P. Chen, U. Villa, and O. Ghattas, “Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems,” Computer Methods in Applied Mechanics and Engineering , vol. 327, pp. 147–172, 2017. [Online]. Available: https://doi.org/10.1016/j.cma.2017.08.016
2017
Cited alongside, same era.
B. Crestel, A. Alexanderian, G. Stadler, and O. Ghattas, “A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem,” Inverse Problems , vol. 33, no. 7, p. 074008, 2017. [Online]. Available: http://iopscience.iop.org/10.1088/1361-6420/aa6d8e
2017
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W. M. Czarnecki, S. Osindero, M. Jaderberg, G. Swirszcz, and R. Pascanu, “Sobolev training for neural networks,” Advances in Neural Information Processing Systems , vol. 30, 2017
2017
Cited alongside, same era.
P. Chen and O. Ghattas, “Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters,” SIAM/ASA Journal on Uncertainty Quantification , vol. 9, no. 4, pp. 1381–1410, 2021
2021
Later among the works it cites.
O. Ghattas and K. Willcox, “Learning physics-based models from data: perspectives from inverse problems and model reduction,” Acta Numerica , vol. 30, pp. 445–554, 2021
2021
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P. Chen, M. Haberman, and O. Ghattas, “Optimal design of acoustic metamaterial cloaks under uncertainty,” Journal of Computational Physics , vol. 431, p. 110114, 2021
2021
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2021
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N. H. Nelsen and A. M. Stuart, “The random feature model for input-output maps between banach spaces,” SIAM Journal on Scientific Computing 43 (5), A3212-A3243 , 2021
2021
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T. O’Leary-Roseberry and U. Villa, hippyflow: Dimension reduced surrogate construction for parametric PDE maps in Python , 2021. [Online]. Available: https://github.com/hippylib/hippyflow
2021
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S. Fresca and A. Manzoni, “POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition,” Computer Methods in Applied Mechanics and Engineering , vol. 388, p. 114181, 2022
2022
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T. O’Leary-Roseberry, U. Villa, P. Chen, and O. Ghattas, “Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs,” Computer Methods in Applied Mechanics and Engineering , vol. 388, p. 114199, 2022
2022
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T. O’Leary-Roseberry, X. Du, A. Chaudhuri, J. R. Martins, K. Willcox, and O. Ghattas, “Learning high-dimensional parametric maps via reduced basis adaptive residual networks,” Computer Methods in Applied Mechanics and Engineering , vol. 402, p. 115730, 2022
2022
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2022
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2022
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2022
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H. Zhang, E. M. Constantinescu, and B. F. Smith, “PETSc TSAdjoint: a discrete adjoint ode solver for first-order and second-order sensitivity analysis,” SIAM Journal on Scientific Computing , vol. 44, no. 1, pp. C1–C24, 2022
2022
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N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandkumar, “Neural operator: Learning maps between function spaces with applications to pdes,” Journal of Machine Learning Research , vol. 24, no. 89, pp. 1–97, 2023
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2023
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T. O’Leary-Roseberry, dino: Derivative-informed neural operator, an efficient framework for high-dimensional parametric derivative learning , v0.2.0 ed., 2023. [Online]. Available: https://github.com/tomoleary/dino
2023
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L. Cao, T. O’Leary-Roseberry, P. K. Jha, J. T. Oden, and O. Ghattas, “Residual-based error correction for neural operator accelerated infinite-dimensional bayesian inverse problems,” Journal of Computational Physics , vol. 486, p. 112104, 2023
2023
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K. Kim, U. Villa, M. Parno, Y. Marzouk, O. Ghattas, and N. Petra, “hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty,” ACM Transactions on Mathematical Software , 2023, to appear
2023
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