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A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions.
Imaging the earth’s interior , volume 1
Claerbout, J. F · 1985
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Reconstructions from boundary measurements
Nachman, A · 1988
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Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography
Isaacson, D., Mueller, J. L., Newell, J. C., and Siltanen, S · 2004
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Iterative methods for approximate solution of inverse problems , volume 577
Bakushinsky, A. B. and Kokurin, M. Y · 2005
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Inverse problem theory and methods for model parameter estimation
Tarantola, A · 2005
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Stability estimates in stationary inverse transport
Bal, G. and Jollivet, A · 2008
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The seismic reflection inverse problem
Symes, W. W · 2009
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Electrical impedance tomography and Calderón’s problem
Uhlmann, G · 2009
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Nonlinear least squares for inverse problems: theoretical foundations and step-by-step guide for applications
Chavent, G · 2010
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Stability of Calderón’s inverse conductivity problem in the plane for discontinuous conductivities
Clop, A., Faraco, D., and Ruiz, A · 2010
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Partial differential equations , volume 19
Evans, L. C · 2010
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Inverse problems: a Bayesian perspective
Stuart, A. M · 2010
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Seismic Data Analysis
Yilmaz, O · 2011
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Linear and nonlinear inverse problems with practical applications
Muller, J. and Siltanen, S · 2012
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Deep learning
Goodfellow, I., Bengio, Y., and Courville, A · 2016
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A Lipschitz stable reconstruction formula for the inverse problem for the wave equation
Liu, S. and Oksanen, L · 2016
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On the stable recovery of a metric from the hyperbolic DN map with incomplete data
Stefanov, P., Uhlmann, G., and Vasy, A · 2016
Cited alongside, same era.
DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E · 2021
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OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion
Deng, C., Feng, S., Wang, H., Zhang, X., Jin, P., Feng, Y., Zeng, Q., Chen, Y., and Lin, Y · 2021
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Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
Lu, L., Jin, P., Pang, G., Zhang, Z., and Karniadakis, G. E · 2021
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MAgNet: Mesh Agnostic Neural PDE solver
Boussif, O., Assouline, D., Benabbou, L., and Bengio, Y · 2022
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Isakov, V · 2017
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Scattering control for the wave equation with unknown wave speed
Caday, P., de Hoop, M. V., Katsnelson, V., and Uhlmann, G · 2019
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Inverse problems for the stationary transport equation in the diffusion scaling
Lai, R.-Y., Li, Q., and Uhlmann, G · 2019
Cited alongside, same era.
Mao, Z., Lu, L., Marxen, O., Zaki, T., and Karniadakis, G. E · 2020
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On universal approximation and error bounds for Fourier Neural Operators
Kovachki, N., Lanthaler, S., and Mishra, S
Cited in the paper.
Neural operator: Learning maps between function spaces
Kovachki, N., Li, Z., Liu, B., Azizzadensheli, K., Bhattacharya, K., Stuart, A., and Anandkumar, A
Cited in the paper.
Brandstetter, J., Worrall, D. E., and Welling, M · 2022
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Learning operators with coupled attention
Kissas, G., Seidman, J. H., Guilhoto, L. F., Preciado, V. M., Pappas, G. J., and Perdikaris, P · 2022
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Error estimates for DeepONets: A deep learning framework in infinite dimensions
Lanthaler, S., Mishra, S., and Karniadakis, G. E · 2022
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Deep learning architectures for nonlinear operator functions and nonlinear inverse problems
Maarten, V., Lassas, M., and Wong, C. A · 2022
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Pathak, J., Subramanian, S., Harrington, P., Raja, S., Chattopadhyay, A., Mardani, M., Kurth, T., Hall, D., Li, Z., Azizzadenesheli, K., Hassanzadeh, p., Kashinath, K., and Anandkumar, A · 2022
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Variable input deep operator networks
Prasthofer, M., De Ryck, T., and Mishra, S · 2022
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