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arxiv: 2412.07010 · v2 · submitted 2024-12-09 · 💻 cs.LG · physics.comp-ph

TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems

Hai V. Nguyen , Tan Bui-Thanh , Clint Dawson This is my paper

Pith reviewed 2026-05-23 07:29 UTC · model grok-4.3

classification 💻 cs.LG physics.comp-ph
keywords Tikhonov autoencodermodel-constrained learninginverse problemssurrogate modelsdata randomizationforward problemsscarce data
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The pith

A Tikhonov autoencoder learns accurate forward and inverse surrogates from one observation sample.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a model-constrained Tikhonov autoencoder can train both forward and inverse surrogate models using only a single arbitrary observation. The central innovation is a data randomization strategy that generates varied instances to explore the solution space and enforce regularization during learning. This matters for applications where collecting large datasets is impractical yet fast, reliable solutions to forward and inverse problems are needed. The approach supplies error bounds for the linear case and demonstrates performance on par with classical Tikhonov and numerical solvers on nonlinear test problems while running orders of magnitude faster.

Core claim

The TAE framework learns both forward and inverse surrogate models from a single arbitrary observation sample. Theoretical error bounds are derived for linear forward and inverse inference by comparing equivalent formulations against pure data-driven and model-constrained baselines. The data randomization strategy serves as a generative mechanism that explores the training space sufficiently to regularize learning. Experiments on 2D heat conductivity inversion and time-dependent 2D Navier-Stokes initial-condition reconstruction show accuracy comparable to traditional Tikhonov solvers and numerical forward solvers together with substantial computational speedups.

What carries the argument

Tikhonov autoencoder whose loss incorporates the forward model operator together with a data randomization strategy that generates multiple consistent training pairs from one observation.

If this is right

  • TAE matches the accuracy of classical Tikhonov solvers on inverse problems while running orders of magnitude faster.
  • TAE matches numerical forward solvers on forward problems at similar speed gains.
  • The same trained network supplies both forward and inverse surrogates.
  • Error bounds hold for linear problems under the derived equivalence to model-constrained Tikhonov regularization.
  • The framework extends to nonlinear cases without requiring large training sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The single-sample regime could reduce the cost of repeated inverse solves in engineering design loops where each new observation is expensive to obtain.
  • Theoretical analysis for nonlinear problems would strengthen the method if the randomization strategy can be shown to control generalization error beyond the linear bounds.
  • Embedding additional physical constraints beyond the forward operator might further stabilize training when the single observation lies far from the training distribution.

Load-bearing premise

The data randomization strategy sufficiently explores the data space to regularize learning and avoid overfitting for both linear and nonlinear problems.

What would settle it

Run TAE on a fresh nonlinear inverse problem using one observation and compare the recovered solution error against a standard Tikhonov solver applied to the same observation; a large gap in accuracy would refute the central performance claim.

Figures

Figures reproduced from arXiv: 2412.07010 by Clint Dawson, Hai V. Nguyen, Tan Bui-Thanh.

Figure 1
Figure 1. Figure 1: The schematic of TAEN approach. A sequential learning strategy is applied to learn the encoder and decoder in two phases. In Phase 1, at every epoch during training, we randomize the observation data with noise ε ∼ N  0, ε2 [diag (y)]2  which is added to the observation data y to generate randomized observation samples. The randomized data is then fed into the encoder network Ψe to predict the inverse so… view at source ↗
Figure 2
Figure 2. Figure 2: 2D heat equation. Left: Domain, boundary conditions, 16 × 16 finite element dis￾cretization mesh, and 10 random observation locations. Middle: A sample of the PoI (the heat conductivity field). Right: The corresponding state (temperature field), observations (tempera￾tures) are taken at 10 observed points. This pair of PoI and observation sample is used for training in one training sample case [PITH_FULL_… view at source ↗
Figure 3
Figure 3. Figure 3: 2D heat equation. Mean and standard deviation of absolute error for 500 test inverse solutions obtained from different approaches. Black points are observational locations. Note that TAEN and TAEN-Full (and similarly for nPOP and mcPOP approaches) have the same encoder (that encodes the inverse solutions), their (identical) results are shown on the 5th row. Relatively to the Tikhonov approach (Tik), the mo… view at source ↗
Figure 4
Figure 4. Figure 4: 2D heat equation. The comparison of 500 test predicted forward solution (at the observational locations) obtained from different approaches. In all plots plot, the x-axis is the magnitude of the true observation, and the y-axis is the magnitude of the predicted observation, both axises has range of [0, 3]. The red line indicates the perfect matching between predictions and truth observations. Top row: Trai… view at source ↗
Figure 5
Figure 5. Figure 5: 2D heat equation. Mean and standard deviation of absolute pointwise error for 500 full state test solutions obtained from TAEN-Full and mcOPO-Full. Black dots are the observational locations. The former is more accurate, especially for the case with one training sample in which it achieves two orders of magnitude smaller error. term. Next, we provide further details on learning PtO/forward maps (see table … view at source ↗
Figure 6
Figure 6. Figure 6: 2D heat equation. A (random) representative case of inverse and full forward solution obtained by TAEN-Full trained with 1 training sample coupled with data randomization of noise level σ = 0.1. TAEN-Full inverse solution is comparable to the Tikhonov (Tik) inverse counterpart, and both are consistent with the ground truth (True). TAEN-Full full forward solution is almost identical (in fact within 3 digits… view at source ↗
Figure 7
Figure 7. Figure 7: 2D heat equation. Relative error of inverse solution over 500 test samples with different noise levels. sults provide additional validation of the TAEN-Full framework’s efficacy in learning forward mappings (in tandem with learning the inverse solutions). We have also seen that, for the larger data set of 100 samples, the accuracy of for￾26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 2D heat equation. Left: Index of 10 observational locations. Right: Mean and standard deviation of observation magnitudes of 10000 true observation samples at the observa￾tional locations. The magnitudes of the predicted solutions of 10 different observation samples for single-sample training cases. ward and inverse maps for all approaches is improved as expected. TAEN and TAEN-Full maintain their superior… view at source ↗
Figure 9
Figure 9. Figure 9: 2D Navier–Stokes equation. Left: A sample of the PoI u. Right: A corresponding vorticity field ω at final time T = 10, observation y are extracted at 20 random observed points. This pair of PoI and observation/vorticity field is used for training in one training sample case. Generating train and test data sets. To generate data pairs of (u, ω), we draw samples of u(x) using the truncated Karhunen-Lo`eve ex… view at source ↗
Figure 10
Figure 10. Figure 10: 2D Navier–Stokes equation. Mean and standard deviation of absolute error of 500 test inverse solutions obtained from different approaches. Black points are observation locations. Relatively to the Tikhonov approach (Tik), the model-constrained approaches are more accurate, and within the model-constrained approaches, TAEN and TAEN-Full are the most accurate ones: in fact one training sample is sufficient … view at source ↗
Figure 11
Figure 11. Figure 11: 2D Navier–Stokes equation. The comparison of the predicted observations on 500 test samples. In all plots plot, the x-axis is the magnitude of the true observation, and the y-axis is the magnitude of the predicted observation, both axes have a range of [−3, 3]. The red line indicates the perfect matching between predictions and the ground truth observation data set. Top row: Trained with 1 training sample… view at source ↗
Figure 12
Figure 12. Figure 12: 2D Navier–Stokes equation. Mean and standard deviation of absolute pointwise error for 500 test vorticity solutions at T = 10 obtained from mcOPO-Full and TAEN-Full. Black points are observational locations. TAEN-Full is more accurate, especially for the case with one training sample in which it achieves two orders of magnitude smaller error. Learned inverse and PtO/forward maps accuracy. Following the sa… view at source ↗
Figure 13
Figure 13. Figure 13: 2D Navier–Stokes equation. A (random) representative case of inverse a and full forward solution at T = 10 obtained by TAEN-Full trained with 1 training sample coupled with data randomization of noise level σ = 0.25 TAEN-Full inverse solution is comparable to the Tikhonov (Tik) inverse counterpart, and both are consistent with the ground truth (True). TAEN-Full full forward solution is almost identical (i… view at source ↗
Figure 14
Figure 14. Figure 14: 2D Navier–Stokes equation. Relative error of inverse solution over 500 test samples with different noise levels. TAEN-Full robustness to arbitrary single-sample. The robustness of TAEN-Full to an arbitrary one-training sample is examined. To be more specific, we randomly pick 12 samples out of 100 training sample data sets. The indices of 20 random observation locations are presented in the left figure in… view at source ↗
Figure 15
Figure 15. Figure 15: 2D Navier–Stokes equation. Left: Index of 20 observational locations. Right: Mean and standard deviation of observation magnitudes of 10000 true observation samples at observational locations. The observation magnitudes of 12 different single-sample training cases [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
read the original abstract

Efficient real-time solvers for forward and inverse problems are essential in engineering and science applications. Machine learning surrogate models have emerged as promising alternatives to traditional methods, offering substantially reduced computational time. Nevertheless, these models typically demand extensive training datasets to achieve robust generalization across diverse scenarios. While physics-based approaches can partially mitigate this data dependency and ensure physics-interpretable solutions, addressing scarce data regimes remains a challenge. Both purely data-driven and physics-based machine learning approaches demonstrate severe overfitting issues when trained with insufficient data. We propose a novel Tikhonov autoencoder model-constrained framework, called TAE, capable of learning both forward and inverse surrogate models using a single arbitrary observation sample. We develop comprehensive theoretical foundations including forward and inverse inference error bounds for the proposed approach for linear cases. For comparative analysis, we derive equivalent formulations for pure data-driven and model-constrained approach counterparts. At the heart of our approach is a data randomization strategy, which functions as a generative mechanism for exploring the training data space, enabling effective training of both forward and inverse surrogate models from a single observation, while regularizing the learning process. We validate our approach through extensive numerical experiments on two challenging inverse problems: 2D heat conductivity inversion and initial condition reconstruction for time-dependent 2D Navier-Stokes equations. Results demonstrate that TAE achieves accuracy comparable to traditional Tikhonov solvers and numerical forward solvers for both inverse and forward problems, respectively, while delivering orders of magnitude computational speedups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes the TAE (Tikhonov Autoencoder) framework, which combines a model-constrained Tikhonov regularization with a data randomization strategy to learn both forward and inverse surrogate models from a single arbitrary observation sample. It derives forward and inverse inference error bounds for linear cases, provides equivalent formulations for pure data-driven and model-constrained baselines, and validates the approach on 2D heat conductivity inversion and initial condition reconstruction for time-dependent 2D Navier-Stokes equations, claiming accuracy comparable to traditional Tikhonov and numerical solvers with orders-of-magnitude speedups.

Significance. If the central claims hold, the work would be significant for enabling real-time forward/inverse solvers in data-scarce regimes common to engineering and science applications. The explicit derivation of linear error bounds (with comparisons to baselines) and the single-observation training regime represent clear strengths; the empirical results on a nonlinear PDE problem further suggest practical utility if the randomization strategy generalizes reliably.

major comments (2)
  1. [Theoretical foundations] Theoretical foundations section: error bounds are stated only for linear cases, yet the central claim (and the Navier-Stokes experiment) extends to nonlinear problems. The data randomization strategy is presented as a generative mechanism that regularizes learning, but no theoretical guarantee is provided that it sufficiently explores the space to prevent the overfitting the paper itself identifies in scarce-data baselines.
  2. [Numerical experiments] Numerical experiments / results sections: the manuscript reports comparable accuracy to traditional solvers but provides no error bars, no description of post-training validation procedure, and no quantitative assessment of how the single-observation randomization explores the relevant function space for the nonlinear case. These omissions make it impossible to evaluate whether the reported performance is robust or merely an artifact of the chosen observation.
minor comments (2)
  1. Title uses TAEN while abstract and body consistently use TAE; standardize nomenclature.
  2. Abstract states that equivalent formulations are derived for baselines, but the main text should explicitly reference the corresponding equations or sections for those derivations to allow direct comparison.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments. We address each major point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Theoretical foundations] Theoretical foundations section: error bounds are stated only for linear cases, yet the central claim (and the Navier-Stokes experiment) extends to nonlinear problems. The data randomization strategy is presented as a generative mechanism that regularizes learning, but no theoretical guarantee is provided that it sufficiently explores the space to prevent the overfitting the paper itself identifies in scarce-data baselines.

    Authors: The manuscript explicitly derives forward and inverse error bounds only for linear cases in the theoretical foundations section, with the Navier-Stokes results presented as empirical validation. We agree that no theoretical guarantee is supplied for the data randomization strategy in the nonlinear regime. In revision we will add an explicit statement clarifying the linear scope of the bounds and a limitations paragraph noting that nonlinear performance relies on empirical evidence. revision: partial

  2. Referee: [Numerical experiments] Numerical experiments / results sections: the manuscript reports comparable accuracy to traditional solvers but provides no error bars, no description of post-training validation procedure, and no quantitative assessment of how the single-observation randomization explores the relevant function space for the nonlinear case. These omissions make it impossible to evaluate whether the reported performance is robust or merely an artifact of the chosen observation.

    Authors: We acknowledge these omissions limit assessment of robustness. The revised manuscript will include error bars computed over multiple random seeds, a clear description of the post-training validation procedure, and quantitative metrics (e.g., sample diversity statistics) characterizing how the randomization strategy explores the function space in the nonlinear experiments. revision: yes

standing simulated objections not resolved
  • Providing a rigorous theoretical guarantee that the data randomization strategy prevents overfitting for nonlinear problems.

Circularity Check

0 steps flagged

No significant circularity; error bounds and validation are independent of the method's outputs

full rationale

The paper derives forward and inverse inference error bounds separately for linear cases and validates the TAE framework through numerical experiments on both linear (2D heat conductivity) and nonlinear (Navier-Stokes) problems. The data randomization strategy is presented as an explicit component of the training process rather than a fitted or self-defined quantity. No equations or claims reduce the performance to a prediction that is equivalent to its inputs by construction, nor do self-citations form a load-bearing chain for the central results. The derivation chain remains self-contained against external benchmarks and empirical testing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach depends on the unverified effectiveness of the randomization strategy as a generative mechanism and on the transferability of linear error bounds to the nonlinear test cases; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Data randomization on a single observation generates a sufficiently rich training distribution to regularize both forward and inverse learning without bias.
    Abstract states this strategy enables effective training from one sample.
  • domain assumption The model-constrained Tikhonov formulation prevents overfitting in scarce-data regimes for the tested inverse problems.
    Central to the claim that the method works where pure data-driven approaches fail.

pith-pipeline@v0.9.0 · 5802 in / 1352 out tokens · 42726 ms · 2026-05-23T07:29:18.530076+00:00 · methodology

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Reference graph

Works this paper leans on

97 extracted references · 97 canonical work pages · 4 internal anchors

  1. [1]

    Modl: Model-based deep learning architec- ture for inverse problems

    Hemant K Aggarwal, Merry P Mani, and Mathews Jacob. Modl: Model-based deep learning architec- ture for inverse problems. IEEE transactions on medical imaging , 38(2):394–405, 2018

    work page 2018

  2. [2]

    Variational autoencoder inverse mapper: An end-to-end deep learning framework for inverse problems

    Manal Almaeen, Yasir Alanazi, Nobuo Sato, W Melnitchouk, Michelle P Kuchera, and Yaohang Li. Variational autoencoder inverse mapper: An end-to-end deep learning framework for inverse problems. In 2021 International Joint Conference on Neural Networks (IJCNN) , pages 1–8. IEEE, 2021

    work page 2021

  3. [3]

    Solving inverse problems using data-driven models

    Simon Arridge, Peter Maass, Ozan ¨Oktem, and Carola-Bibiane Sch¨ onlieb. Solving inverse problems using data-driven models. Acta Numerica, 28:1–174, 2019

    work page 2019

  4. [4]

    Architecting smart city digital twins: Combined semantic model and machine learning approach

    Mark Austin, Parastoo Delgoshaei, Maria Coelho, and Mohammad Heidarinejad. Architecting smart city digital twins: Combined semantic model and machine learning approach. Journal of Management in Engineering, 36(4):04020026, 2020

    work page 2020

  5. [5]

    Interpretable fine-tuning for graph neural network surrogate models

    Shivam Barwey and Romit Maulik. Interpretable fine-tuning for graph neural network surrogate models. arXiv preprint arXiv:2311.07548 , 2023

    work page arXiv 2023

  6. [6]

    Neural network augmented inverse problems for PDEs

    Jens Berg and Kaj Nystr¨ om. Neural network augmented inverse problems for pdes. arXiv preprint arXiv:1712.09685, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    Training with noise is equivalent to tikhonov regularization

    Chris M Bishop. Training with noise is equivalent to tikhonov regularization. Neural computation, 7(1):108–116, 1995

    work page 1995

  8. [8]

    Learned svd: solving inverse problems via hybrid autoencoding

    Yoeri E Boink and Christoph Brune. Learned svd: solving inverse problems via hybrid autoencoding. arXiv preprint arXiv:1912.10840 , 2019. 50

    work page arXiv 1912

  9. [9]

    JAX: composable transformations of Python+NumPy programs, 2018

    James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018

    work page 2018

  10. [10]

    Analysis of the Hessian for inverse scattering problems

    Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves. Inverse Problems , 28(5):055001, 2012. http://users.ices.utexas.edu/%7Etanbui/PublishedPapers/CompactI.pdf

    work page 2012

  11. [11]

    Analysis of the Hessian for inverse scattering problems

    Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves. Inverse Problems , 28(5):055002, 2012. http://users.ices.utexas.edu/%7Etanbui/PublishedPapers/CompactII.pdf

    work page 2012

  12. [12]

    Analysis of the Hessian for inverse scattering problems

    Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves. Inverse Problems and Imaging, 2013. http://users.ices.utexas.edu/%7Etanbui/PublishedPapers/EM3Dmedium.pdf

    work page 2013

  13. [13]

    Physics- informed neural networks for heat transfer problems

    Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Karniadakis. Physics- informed neural networks for heat transfer problems. Journal of Heat Transfer, 143(6):060801, 2021

    work page 2021

  14. [14]

    Deep feature learning for medical image analysis with convolutional autoencoder neural network.IEEE Transactions on Big Data, 7(4):750–758, 2017

    Min Chen, Xiaobo Shi, Yin Zhang, Di Wu, and Mohsen Guizani. Deep feature learning for medical image analysis with convolutional autoencoder neural network.IEEE Transactions on Big Data, 7(4):750–758, 2017

    work page 2017

  15. [15]

    Using ma- chine learning to support qualitative coding in social science: Shifting the focus to ambiguity

    Nan-Chen Chen, Margaret Drouhard, Rafal Kocielnik, Jina Suh, and Cecilia R Aragon. Using ma- chine learning to support qualitative coding in social science: Shifting the focus to ambiguity. ACM Transactions on Interactive Intelligent Systems (TiiS) , 8(2):1–20, 2018

    work page 2018

  16. [16]

    Neural ordinary differ- ential equations

    Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differ- ential equations. Advances in neural information processing systems , 31, 2018

    work page 2018

  17. [17]

    Physics-informed neural networks for inverse problems in nano-optics and metamaterials

    Yuyao Chen, Lu Lu, George Em Karniadakis, and Luca Dal Negro. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics express, 28(8):11618–11633, 2020

    work page 2020

  18. [18]

    Paired autoencoders for inverse problems

    Matthias Chung, Emma Hart, Julianne Chung, Bas Peters, and Eldad Haber. Paired autoencoders for inverse problems. arXiv preprint arXiv:2405.13220 , 2024

    work page arXiv 2024

  19. [19]

    Accelerating markov chain monte carlo with active subspaces

    Paul G Constantine, Carson Kent, and Tan Bui-Thanh. Accelerating markov chain monte carlo with active subspaces. SIAM Journal on Scientific Computing , 38(5):A2779–A2805, 2016

    work page 2016

  20. [20]

    Scientific machine learning through physics–informed neural networks: Where we are and what’s next

    Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing , 92(3):88, 2022

    work page 2022

  21. [21]

    Imagenet: A large-scale hierarchical image database

    Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition , pages 248–255. Ieee, 2009

    work page 2009

  22. [22]

    Improving generalization performance using double backpropagation

    Harris Drucker and Yann Le Cun. Improving generalization performance using double backpropagation. IEEE transactions on neural networks , 3(6):991–997, 1992

    work page 1992

  23. [23]

    Finite volume methods.Handbook of numer- ical analysis, 7:713–1018, 2000

    Robert Eymard, Thierry Gallou¨ et, and Rapha` ele Herbin. Finite volume methods.Handbook of numer- ical analysis, 7:713–1018, 2000

    work page 2000

  24. [24]

    Solving inverse problems in steady-state navier- stokes equations using deep neural networks

    Tiffany Fan, Kailai Xu, Jay Pathak, and Eric Darve. Solving inverse problems in steady-state navier- stokes equations using deep neural networks. arXiv preprint arXiv:2008.13074 , 2020

    work page arXiv 2008

  25. [25]

    The rank of a random matrix

    Xinlong Feng and Zhinan Zhang. The rank of a random matrix. Applied Mathematics and Computation, 185(1):689–694, 2007

    work page 2007

  26. [26]

    Scaleable input gradient regularization for adversarial robustness

    Chris Finlay and Adam M Oberman. Scaleable input gradient regularization for adversarial robustness. Machine Learning with Applications , 3:100017, 2021

    work page 2021

  27. [27]

    Approximation of dynamical systems by continuous time recurrent neural networks

    Ken-ichi Funahashi and Yuichi Nakamura. Approximation of dynamical systems by continuous time recurrent neural networks. Neural networks, 6(6):801–806, 1993

    work page 1993

  28. [28]

    Transformers for modeling physical systems

    Nicholas Geneva and Nicholas Zabaras. Transformers for modeling physical systems. Neural Networks, 146:272–289, 2022

    work page 2022

  29. [29]

    Finite difference method for numerical computation of discon- tinuous solutions of the equations of fluid dynamics

    Sergei K Godunov and I Bohachevsky. Finite difference method for numerical computation of discon- tinuous solutions of the equations of fluid dynamics. Matematiˇ ceskij sbornik, 47(3):271–306, 1959

    work page 1959

  30. [30]

    Solving bayesian inverse 51 problems via variational autoencoders

    Hwan Goh, Sheroze Sheriffdeen, Jonathan Wittmer, and Tan Bui-Thanh. Solving bayesian inverse 51 problems via variational autoencoders. arXiv preprint arXiv:1912.04212 , 2019

    work page arXiv 1912

  31. [31]

    Tikhonov regularization and total least squares

    Gene H Golub, Per Christian Hansen, and Dianne P O’Leary. Tikhonov regularization and total least squares. SIAM journal on matrix analysis and applications , 21(1):185–194, 1999

    work page 1999

  32. [32]

    Medical image denoising using convolutional denoising autoencoders

    Lovedeep Gondara. Medical image denoising using convolutional denoising autoencoders. In 2016 IEEE 16th international conference on data mining workshops (ICDMW) , pages 241–246. IEEE, 2016

    work page 2016

  33. [33]

    Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems

    Francisco J Gonzalez and Maciej Balajewicz. Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. arXiv preprint arXiv:1808.01346 , 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [34]

    Partial differential equations of mathematical physics and integral equations

    Ronald B Guenther and John W Lee. Partial differential equations of mathematical physics and integral equations. Courier Corporation, 1996

    work page 1996

  35. [35]

    Masked autoen- coders are scalable vision learners

    Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Doll´ ar, and Ross Girshick. Masked autoen- coders are scalable vision learners. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 16000–16009, 2022

    work page 2022

  36. [36]

    Inverse problems: Tikhonov theory and algorithms , volume 22

    Kazufumi Ito and Bangti Jin. Inverse problems: Tikhonov theory and algorithms , volume 22. World Scientific, 2014

    work page 2014

  37. [37]

    Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems

    Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering , 365:113028, 2020

    work page 2020

  38. [38]

    Physics-informed neural networks for inverse problems in supersonic flows

    Ameya D Jagtap, Zhiping Mao, Nikolaus Adams, and George Em Karniadakis. Physics-informed neural networks for inverse problems in supersonic flows. Journal of Computational Physics , 466:111402, 2022

    work page 2022

  39. [39]

    A physics-driven deep-learning network for solving nonlinear inverse problems

    Yuchen Jin, Qiuyang Shen, Xuqing Wu, Jiefu Chen, and Yueqin Huang. A physics-driven deep-learning network for solving nonlinear inverse problems. Petrophysics, 61(01):86–98, 2020

    work page 2020

  40. [40]

    Deep-learning-based surrogate model for reservoir simulation with time-varying well controls

    Zhaoyang Larry Jin, Yimin Liu, and Louis J Durlofsky. Deep-learning-based surrogate model for reservoir simulation with time-varying well controls. Journal of Petroleum Science and Engineering , 192:107273, 2020

    work page 2020

  41. [41]

    Statistical and computational inverse problems, volume 160

    Jari Kaipio and Erkki Somersalo. Statistical and computational inverse problems, volume 160. Springer Science & Business Media, 2006

    work page 2006

  42. [42]

    Adam: A Method for Stochastic Optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  43. [43]

    Imagenet classification with deep convolutional neural networks

    Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems , 25, 2012

    work page 2012

  44. [44]

    Diagonal recurrent neural networks for dynamic systems control

    Chao-Chee Ku and Kwang Y Lee. Diagonal recurrent neural networks for dynamic systems control. IEEE transactions on neural networks , 6(1):144–156, 1995

    work page 1995

  45. [45]

    Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

    Kookjin Lee and Kevin T Carlberg. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. Journal of Computational Physics , 404:108973, 2020

    work page 2020

  46. [46]

    Finite volume methods for hyperbolic problems , volume 31

    Randall J LeVeque. Finite volume methods for hyperbolic problems , volume 31. Cambridge university press, 2002

    work page 2002

  47. [47]

    Nett: Solving inverse prob- lems with deep neural networks

    Housen Li, Johannes Schwab, Stephan Antholzer, and Markus Haltmeier. Nett: Solving inverse prob- lems with deep neural networks. Inverse Problems, 36(6):065005, 2020

    work page 2020

  48. [48]

    Transformer for partial differential equations’ operator learning

    Zijie Li, Kazem Meidani, and Amir Barati Farimani. Transformer for partial differential equations’ operator learning. arXiv preprint arXiv:2205.13671 , 2022

    work page arXiv 2022

  49. [49]

    Fourier Neural Operator for Parametric Partial Differential Equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895 , 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  50. [50]

    Fourier neural operator for parametric partial differential equations, 2020

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations, 2020

    work page 2020

  51. [51]

    Training deep neural networks for the inverse design of nanophotonic structures

    Dianjing Liu, Yixuan Tan, Erfan Khoram, and Zongfu Yu. Training deep neural networks for the inverse design of nanophotonic structures. Acs Photonics, 5(4):1365–1369, 2018

    work page 2018

  52. [52]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218–229, 2021. 52

    work page 2021

  53. [53]

    Adversarial regularizers in inverse prob- lems

    Sebastian Lunz, Ozan ¨Oktem, and Carola-Bibiane Sch¨ onlieb. Adversarial regularizers in inverse prob- lems. Advances in neural information processing systems , 31, 2018

    work page 2018

  54. [54]

    Noise injection into inputs in back-propagation learning

    Kiyotoshi Matsuoka. Noise injection into inputs in back-propagation learning. IEEE Transactions on Systems, Man, and Cybernetics , 22(3):436–440, 1992

    work page 1992

  55. [55]

    Multistep prediction of dynamic systems with recurrent neural networks

    Nima Mohajerin and Steven L Waslander. Multistep prediction of dynamic systems with recurrent neural networks. IEEE transactions on neural networks and learning systems , 30(11):3370–3383, 2019

    work page 2019

  56. [56]

    Digital twins that learn and correct themselves

    Beatriz Moya, Alberto Bad´ ıas, Ic´ ıar Alfaro, Francisco Chinesta, and El´ ıas Cueto. Digital twins that learn and correct themselves. International Journal for Numerical Methods in Engineering , 123(13):3034–3044, 2022

    work page 2022

  57. [57]

    Dias: a data-informed active subspace regular- ization framework for inverse problems

    Hai Nguyen, Jonathan Wittmer, and Tan Bui-Thanh. Dias: a data-informed active subspace regular- ization framework for inverse problems. Computation, 10(3):38, 2022

    work page 2022

  58. [58]

    A model-constrained tangent slope learning approach for dynamical systems

    Hai V Nguyen and Tan Bui-Thanh. A model-constrained tangent slope learning approach for dynamical systems. International Journal of Computational Fluid Dynamics , 36(7):655–685, 2022

    work page 2022

  59. [59]

    Tnet: A model-constrained tikhonov network approach for inverse problems

    Hai V Nguyen and Tan Bui-Thanh. Tnet: A model-constrained tikhonov network approach for inverse problems. SIAM Journal on Scientific Computing , 46(1):C77–C100, 2024

    work page 2024

  60. [60]

    Numerical optimization

    Jorge Nocedal and Stephen J Wright. Numerical optimization. Springer, 1999

    work page 1999

  61. [61]

    Derivative-informed neu- ral operator: an efficient framework for high-dimensional parametric derivative learning

    Thomas O’Leary-Roseberry, Peng Chen, Umberto Villa, and Omar Ghattas. Derivative-informed neu- ral operator: an efficient framework for high-dimensional parametric derivative learning. Journal of Computational Physics, 496:112555, 2024

    work page 2024

  62. [62]

    Deep learning techniques for inverse problems in imaging

    Gregory Ongie, Ajil Jalal, Christopher A Metzler, Richard G Baraniuk, Alexandros G Dimakis, and Rebecca Willett. Deep learning techniques for inverse problems in imaging. IEEE Journal on Selected Areas in Information Theory , 1(1):39–56, 2020

    work page 2020

  63. [63]

    CRC press, 2017

    M Necati ¨Ozi¸ sik, Helcio RB Orlande, Marcelo J Cola¸ co, and Renato M Cotta.Finite difference methods in heat transfer. CRC press, 2017

    work page 2017

  64. [64]

    Derivative-informed pro- jected neural networks for high-dimensional parametric maps governed by pdes

    Thomas O’Leary-Roseberry, Umberto Villa, Peng Chen, and Omar Ghattas. Derivative-informed pro- jected neural networks for high-dimensional parametric maps governed by pdes. Computer Methods in Applied Mechanics and Engineering , 388:114199, 2022

    work page 2022

  65. [65]

    Solving inverse-pde problems with physics-aware neural networks

    Samira Pakravan, Pouria A Mistani, Miguel A Aragon-Calvo, and Frederic Gibou. Solving inverse-pde problems with physics-aware neural networks. Journal of Computational Physics , 440:110414, 2021

    work page 2021

  66. [66]

    Long-time predictive modeling of nonlinear dynamical systems using neural networks

    Shaowu Pan and Karthik Duraisamy. Long-time predictive modeling of nonlinear dynamical systems using neural networks. Complexity, 2018, 2018

    work page 2018

  67. [67]

    Geophysical inverse theory, volume 1

    Robert L Parker. Geophysical inverse theory, volume 1. Princeton university press, 1994

    work page 1994

  68. [68]

    A general finite difference method for arbitrary meshes

    Nicholas Perrone and Robert Kao. A general finite difference method for arbitrary meshes. Computers & Structures, 5(1):45–57, 1975

    work page 1975

  69. [69]

    A graph convolutional autoencoder approach to model order reduction for parametrized pdes

    Federico Pichi, Beatriz Moya, and Jan S Hesthaven. A graph convolutional autoencoder approach to model order reduction for parametrized pdes. Journal of Computational Physics , 501:112762, 2024

    work page 2024

  70. [70]

    Networks for approximation and learning

    Tomaso Poggio and Federico Girosi. Networks for approximation and learning. Proceedings of the IEEE, 78(9):1481–1497, 1990

    work page 1990

  71. [71]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics , 378:686–707, 2019

    work page 2019

  72. [72]

    The finite element method in engineering

    Singiresu S Rao. The finite element method in engineering . Elsevier, 2010

    work page 2010

  73. [73]

    The role of ai, machine learning, and big data in digital twinning: A systematic literature review, challenges, and opportunities

    M Mazhar Rathore, Syed Attique Shah, Dhirendra Shukla, Elmahdi Bentafat, and Spiridon Bakiras. The role of ai, machine learning, and big data in digital twinning: A systematic literature review, challenges, and opportunities. IEEE Access, 9:32030–32052, 2021

    work page 2021

  74. [74]

    An introduction to the finite element method, 1993

    JN Reddy. An introduction to the finite element method, 1993

    work page 1993

  75. [75]

    Regularization using jittered training data

    Russell Reed, Seho Oh, RJ Marks, et al. Regularization using jittered training data. In International Joint Conference on Neural Networks , volume 3, pages 147–152, 1992

    work page 1992

  76. [76]

    The little engine that could: Regularization by denoising (red)

    Yaniv Romano, Michael Elad, and Peyman Milanfar. The little engine that could: Regularization by denoising (red). SIAM Journal on Imaging Sciences , 10(4):1804–1844, 2017

    work page 2017

  77. [77]

    Improving the adversarial robustness and interpretability of 53 deep neural networks by regularizing their input gradients

    Andrew Ross and Finale Doshi-Velez. Improving the adversarial robustness and interpretability of 53 deep neural networks by regularizing their input gradients. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018

    work page 2018

  78. [78]

    Learning to simulate complex physics with graph networks

    Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter Battaglia. Learning to simulate complex physics with graph networks. In International Conference on Machine Learning, pages 8459–8468. PMLR, 2020

    work page 2020

  79. [79]

    Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils

    Anand Pratap Singh, Shivaji Medida, and Karthik Duraisamy. Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils. AIAA journal, 55(7):2215–2227, 2017

    work page 2017

  80. [80]

    Implicit neural representations with periodic activation functions

    Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. Advances in neural information processing systems, 33:7462–7473, 2020

    work page 2020

Showing first 80 references.