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arxiv: 2605.16966 · v1 · pith:CJP6YGLWnew · submitted 2026-05-16 · 💻 cs.AI

Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects

Zhentao Tan , Yuze Hao , Boyi Zou , Mingsheng Long , Yi Yang , Gang Bao This is my paper

Pith reviewed 2026-05-19 20:24 UTC · model grok-4.3

classification 💻 cs.AI
keywords inverse PDE problemsartificial intelligencemachine learninginverse designcontrol problemsphysics-informed neural networksscientific computingPDE applications
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The pith

AI methods are reshaping inverse PDE problems by organizing them into unified categories of inference, design, and control.

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

The paper reviews how artificial intelligence addresses inverse partial differential equation problems across scientific domains. It begins with basic formulations, key challenges, and traditional numerical methods before structuring the field into three categories: inverse problems, inverse design, and control problems. Within each category the authors outline methodological paradigms and survey recent state-of-the-art learning-based approaches. Representative applications in mechanical systems, aerodynamics, thermal systems, full-waveform inversion, system identification, and medical imaging are summarized, followed by discussion of open challenges such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. A sympathetic reader cares because these inverse tasks underpin practical technologies in medicine, engineering, and materials science where classical solvers encounter severe limitations.

Core claim

This survey provides the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems through a three-category organization of methodological paradigms and representative state-of-the-art approaches.

What carries the argument

The three-category organization into inverse problems, inverse design, and control problems, which structures the review of learning-based methods for PDE-governed systems.

If this is right

  • AI approaches improve solutions for inverse problems in medical imaging and geophysics.
  • Inverse design tasks in materials science and aerodynamics gain from the reviewed learning paradigms.
  • Control problems in mechanical and thermal systems become more tractable with modern methods.
  • Future prospects include physics-informed architectures and inverse foundation models.
  • Uncertainty quantification and handling of limited real-world data remain key open directions.

Where Pith is reading between the lines

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

  • The three-category structure could guide development of hybrid numerical-AI solvers that inherit strengths from both traditions.
  • Similar categorization might be applied to inverse problems governed by integral equations or stochastic PDEs.
  • Standardized benchmarks derived from the reviewed applications could accelerate progress across the field.
  • The survey's emphasis on foundation models suggests potential transfer learning from large PDE datasets to new physical domains.

Load-bearing premise

The assumption that the chosen methodological paradigms and representative state-of-the-art approaches from recent years, along with the three-category organization, sufficiently capture and structure the full breadth of advances in the field.

What would settle it

Discovery of a significant recent AI advance for an inverse PDE problem that cannot be placed into any of the three categories or that is omitted from the reviewed state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2605.16966 by Boyi Zou, Gang Bao, Mingsheng Long, Yi Yang, Yuze Hao, Zhentao Tan.

Figure 1
Figure 1. Figure 1: Overview of AI-based inverse PDE problems. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Taxonomy of AI-based methods for inverse PDE problems, organized along three main axes: inverse problems, inverse design, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overview of general AI methods for inverse problems, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overview of general AI methods for inverse design. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Overview of general AI methods for control problems. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state-of-the-art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full-waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems.

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 manuscript is a survey on AI methods for inverse PDE problems. It introduces basic formulations, key challenges, and traditional numerical foundations, then organizes advances into three categories—inverse problems, inverse design, and control problems. For each category it reviews methodological paradigms and representative SOTA approaches from recent years, summarizes applications across domains such as mechanical systems, aerodynamics, thermal systems, full-waveform inversion, system identification, and medical imaging, and concludes with open challenges and prospects including physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. The paper claims to deliver the first unified and systematic perspective on the topic.

Significance. If the three-category taxonomy is demonstrated to be principled, non-redundant, and comprehensive, the survey could serve as a valuable unifying reference in scientific machine learning. It would help organize the rapidly growing literature on learning-based methods for inference, design, and control in PDE-governed systems and highlight promising directions such as inverse foundation models.

major comments (2)
  1. [Abstract and Introduction] Abstract and Introduction: The claim to provide the 'first unified and systematic perspective' depends on the three-category taxonomy (inverse problems, inverse design, control problems) being principled rather than arbitrary. The manuscript does not explicitly justify the separation of inverse design from control problems or address potential overlaps (e.g., both involve PDE-constrained optimization but may differ in objectives, data regimes, or loss formulations). Without this justification, hybrid methods risk being forced into one bin, undermining the systematic organization.
  2. [§3 (Organization of the Survey)] §3 (Organization of the Survey): The weakest assumption noted in the review—that the chosen paradigms and three-category structure sufficiently capture the full breadth—requires explicit discussion of coverage, omissions, and alternatives (e.g., classification by method type such as PINNs versus operator learning) to support the central claim.
minor comments (2)
  1. [Applications section] Applications section: Ensure quantitative performance metrics or comparative tables from the cited SOTA works are included where available to move beyond descriptive summaries.
  2. [Future prospects] Future prospects: The discussion of inverse foundation models would benefit from additional concrete references to recent foundation-model efforts in scientific computing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help us clarify the organizational structure of our survey. We address the major comments point by point below, proposing revisions to enhance the justification of our taxonomy.

read point-by-point responses

  1. Referee: [Abstract and Introduction] Abstract and Introduction: The claim to provide the 'first unified and systematic perspective' depends on the three-category taxonomy (inverse problems, inverse design, control problems) being principled rather than arbitrary. The manuscript does not explicitly justify the separation of inverse design from control problems or address potential overlaps (e.g., both involve PDE-constrained optimization but may differ in objectives, data regimes, or loss formulations). Without this justification, hybrid methods risk being forced into one bin, undermining the systematic organization.

    Authors: We acknowledge the importance of justifying the taxonomy to support our claim of providing a unified perspective. In the revised manuscript, we will include an explicit discussion in the Introduction and a new subsection in §3. We will explain that the categories are distinguished primarily by their objectives: inverse problems focus on parameter/state inference from data, inverse design on optimizing static design variables for target outcomes, and control on dynamic steering of system evolution. Although overlaps in PDE-constrained optimization exist, the separation reflects differences in typical problem formulations, data regimes (e.g., static observations vs. time-series), and optimization goals. We will also address hybrid methods by noting how they can be categorized based on the dominant aspect, thus reinforcing the systematic nature of the organization. revision: yes

  2. Referee: [§3 (Organization of the Survey)] §3 (Organization of the Survey): The weakest assumption noted in the review—that the chosen paradigms and three-category structure sufficiently capture the full breadth—requires explicit discussion of coverage, omissions, and alternatives (e.g., classification by method type such as PINNs versus operator learning) to support the central claim.

    Authors: We agree that discussing the rationale and alternatives would strengthen the survey. In the revision, we will expand §3 to explicitly discuss the coverage of the literature, deliberate omissions (such as limiting to AI-centric methods post-2017 while referencing foundational numerical approaches), and alternative classifications, including by methodological paradigms like PINNs, operator learning, or graph neural networks. We will argue that the problem-type based taxonomy (inverse problems, inverse design, control) offers a more coherent view aligned with application domains and practical workflows, compared to method-based ones that may cross-cut categories. This addition will better support the claim of a systematic perspective. revision: yes

Circularity Check

0 steps flagged

No circularity: survey synthesizes external literature without self-referential derivations.

full rationale

This is a review paper whose central contribution is a literature survey and three-category taxonomy (inverse problems, inverse design, control problems) drawn from cited external works. No original equations, predictions, or fitted parameters appear in the abstract or described structure; the claim of providing the 'first unified perspective' rests on the organization of prior results rather than any derivation that reduces to the paper's own inputs by construction. No self-citation load-bearing steps, self-definitional elements, or ansatz smuggling are present. The paper is self-contained against external benchmarks in the sense that its value derives from synthesis of independently published methods, not from internal fitting or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a literature survey paper. It introduces no free parameters, mathematical axioms, or invented entities of its own and instead summarizes prior published research across the cited works.

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Works this paper leans on

278 extracted references · 278 canonical work pages · 10 internal anchors

  1. [1]

    Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. 2016. Tensorflow: Large-scale machine learning on heterogeneous distributed systems.arXiv preprint arXiv:1603.04467(2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  2. [2]

    Robert Acar. 1993. Identification of the coefficient in elliptic equations.SIAM journal on control and optimization31, 5 (1993), 1221–1244

    work page 1993

  3. [3]

    Gabriel Achour, Woong Je Sung, Olivia J Pinon-Fischer, and Dimitri N Mavris. 2020. Development of a conditional generative adversarial network for airfoil shape optimization. InAIAA Scitech 2020 Forum. 2261

    work page 2020

  4. [4]

    Grégoire Allaire, François Jouve, and Anca-Maria Toader. 2004. Structural optimization using sensitivity analysis and a level-set method.Journal of computational physics194, 1 (2004), 363–393

    work page 2004

  5. [5]

    Kelsey Allen, Tatiana Lopez-Guevara, Kimberly L Stachenfeld, Alvaro Sanchez Gonzalez, Peter Battaglia, Jessica B Hamrick, and Tobias Pfaff

    work page

  6. [6]

    Inverse design for fluid-structure interactions using graph network simulators.Advances in neural information processing systems35 (2022), 13759–13774

    work page 2022

  7. [7]

    Sensong An, Clayton Fowler, Bowen Zheng, Mikhail Y Shalaginov, Hong Tang, Hang Li, Li Zhou, Jun Ding, Anuradha Murthy Agarwal, Clara Rivero-Baleine, et al. 2019. A deep learning approach for objective-driven all-dielectric metasurface design.Acs Photonics6, 12 (2019), 3196–3207

    work page 2019

  8. [8]

    Apurva Anand, Koushik Marepally, M Muneeb Safdar, and James D Baeder. 2024. A novel approach to inverse design of wind turbine airfoils using tandem neural networks.Wind Energy27, 9 (2024), 900–921

    work page 2024

  9. [9]

    Lynton Ardizzone, Jakob Kruse, Carsten Rother, and Ullrich Köthe. [n. d.]. Analyzing Inverse Problems with Invertible Neural Networks. In International Conference on Learning Representations

    work page

  10. [10]

    Florian Arnold and Rudibert King. 2021. State-space modeling for control based on physics-informed neural networks.Engineering Applications of Artificial Intelligence101 (2021), 104195

    work page 2021

  11. [11]

    Kai Arulkumaran, Marc Peter Deisenroth, Miles Brundage, and Anil Anthony Bharath. 2017. Deep reinforcement learning: A brief survey.IEEE signal processing magazine34, 6 (2017), 26–38

    work page 2017

  12. [12]

    Yannick Augenstein, Taavi Repan, and Carsten Rockstuhl. 2023. Neural operator-based surrogate solver for free-form electromagnetic inverse design.Acs Photonics10, 5 (2023), 1547–1557

    work page 2023

  13. [13]

    Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, and Anima Anandkumar. 2024. Neural operators for accelerating scientific simulations and design.Nature Reviews Physics6, 5 (2024), 320–328

    work page 2024

  14. [14]

    Michael Bain and Claude Sammut. 1995. A Framework for Behavioural Cloning.. InMachine intelligence 15. 103–129

    work page 1995

  15. [15]

    Kensley Balla, Ruben Sevilla, Oubay Hassan, and Kenneth Morgan. 2022. Inverse Aerodynamic Design Using Neural Networks. InAdvances in Computational Methods and Technologies in Aeronautics and Industry. Springer, 131–143

    work page 2022

  16. [16]

    Arpit Bansal, Hong-Min Chu, Avi Schwarzschild, Soumyadip Sengupta, Micah Goldblum, Jonas Geiping, and Tom Goldstein. 2023. Universal guidance for diffusion models. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition. 843–852

    work page 2023

  17. [17]

    Gang Bao, Shui-Nee Chow, Peijun Li, and Haomin Zhou. 2010. Numerical solution of an inverse medium scattering problem with a stochastic source.Inverse Problems26, 7 (2010), 074014

    work page 2010

  18. [18]

    Gang Bao, Guanghui Hu, Yavar Kian, and Tao Yin. 2018. Inverse source problems in elastodynamics.Inverse Problems34, 4 (2018), 045009

    work page 2018

  19. [19]

    Gang Bao and Peijun Li. 2004. Inverse medium scattering for three-dimensional time harmonic Maxwell equations.Inverse Problems20, 2 (2004), L1–L7

    work page 2004

  20. [20]

    Gang Bao and Peijun Li. 2005. Inverse medium scattering for the Helmholtz equation at fixed frequency.Inverse Problems21, 5 (2005), 1621–1641

    work page 2005

  21. [21]

    2022.Maxwell’s Equations in Periodic Structures

    Gang Bao and Peijun Li. 2022.Maxwell’s Equations in Periodic Structures. Springer

    work page 2022

  22. [22]

    Gang Bao, Peijun Li, Junshan Lin, and Faouzi Triki. 2015. Inverse scattering problems with multi-frequencies.Inverse Problems31, 9 (2015), 093001

    work page 2015

  23. [23]

    Gang Bao and Junshan Lin. 2013. Near-field imaging of the surface displacement on an infinite ground plane.Inverse Problems and Imaging7, 2 (2013), 377–396

    work page 2013

  24. [24]

    Gang Bao, Yuantong Liu, and Faouzi Triki. 2021. Recovering point sources for the inhomogeneous Helmholtz equation.Inverse Problems37, 9 (2021), 095005

    work page 2021

  25. [25]

    Gang Bao, Dong Wang, and Boyi Zou. 2026. Weak adversarial networks for interface optimal design problems under physical constraints.Inverse Problems and Imaging25, 0 (2026), 133–157

    work page 2026

  26. [26]

    Gang Bao, Xiaojing Ye, Yaohua Zang, and Haomin Zhou. 2020. Numerical solution of inverse problems by weak adversarial networks.Inverse Problems36, 11 (2020), 115003

    work page 2020

  27. [27]

    Gang Bao and Kihyun Yun. 2009. On the stability of an inverse problem for the wave equation.Inverse problems25, 4 (2009), 045003

    work page 2009

  28. [28]

    Gang Bao and Yaohua Zang. 2025. ParticleWNN: A weak-from deep learning framework for solving partial differential equations and inverse problems. (2025)

    work page 2025

  29. [29]

    Jostein Barry-Straume, Arash Sarshar, Andrey A Popov, and Adrian Sandu. 2025. Physics-informed neural networks for PDE-constrained optimization and control.Communications on Applied Mathematics and Computation(2025), 1–24. Manuscript submitted to ACM Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects 27

    work page 2025

  30. [30]

    Jan-Hendrik Bastek and Dennis M Kochmann. 2023. Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models. Nature Machine Intelligence5, 12 (2023), 1466–1475

    work page 2023

  31. [31]

    Jens Behrmann, Will Grathwohl, Ricky TQ Chen, David Duvenaud, and Jörn-Henrik Jacobsen. 2019. Invertible residual networks. InInternational conference on machine learning. PMLR, 573–582

    work page 2019

  32. [32]

    Martin Philip Bendsøe and Noboru Kikuchi. 1988. Generating optimal topologies in structural design using a homogenization method.Computer methods in applied mechanics and engineering71, 2 (1988), 197–224

    work page 1988

  33. [33]

    Deniz A Bezgin, Aaron B Buhendwa, and Nikolaus A Adams. 2023. JAX-Fluids: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows.Computer Physics Communications282 (2023), 108527

    work page 2023

  34. [34]

    Luke Bhan, Yuanyuan Shi, and Miroslav Krstic. 2023. Neural operators for bypassing gain and control computations in PDE backstepping.IEEE Trans. Automat. Control69, 8 (2023), 5310–5325

    work page 2023

  35. [35]

    Luke Bhan, Yuanyuan Shi, and Miroslav Krstic. 2023. Operator learning for nonlinear adaptive control. InLearning for dynamics and control conference. PMLR, 346–357

    work page 2023

  36. [36]

    Md Muhtasim Billah, Aminul Islam Khan, Jin Liu, and Prashanta Dutta. 2023. Physics-informed deep neural network for inverse heat transfer problems in materials.Materials Today Communications35 (2023), 106336

    work page 2023

  37. [37]

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

    work page 2018

  38. [38]

    Tan Bui-Thanh, Carsten Burstedde, Omar Ghattas, James Martin, Georg Stadler, and Lucas C Wilcox. 2012. Extreme-scale UQ for Bayesian inverse problems governed by PDEs. InSC’12: Proceedings of the international conference on high performance computing, networking, storage and analysis. IEEE, 1–11

    work page 2012

  39. [39]

    Junhao Cai, Yuji Yang, Weihao Yuan, Yisheng He, Zilong Dong, Liefeng Bo, Hui Cheng, and Qifeng Chen. 2024. Gic: Gaussian-informed continuum for physical property identification and simulation.Advances in Neural Information Processing Systems37 (2024), 75035–75063

    work page 2024

  40. [40]

    Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Karniadakis. 2021. Physics-informed neural networks for heat transfer problems.Journal of Heat Transfer143, 6 (2021), 060801

    work page 2021

  41. [41]

    Horacio M Calderón, Erik Schulz, Thimo Oehlschlägel, and Herbert Werner. 2021. Koopman operator-based model predictive control with recursive online update. In2021 European Control Conference (ECC). IEEE, 1543–1549

    work page 2021

  42. [42]

    Xiang Cao, Qiaoqiao Ding, Xinliang Liu, Lei Zhang, and Xiaoqun Zhang. 2025. Diff-ano: Towards fast high-resolution ultrasound computed tomography via conditional consistency models and adjoint neural operators.arXiv preprint arXiv:2507.16344(2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  43. [43]

    Louis N Cattafesta III and Mark Sheplak. 2011. Actuators for active flow control.Annual Review of Fluid Mechanics43, 1 (2011), 247–272

    work page 2011

  44. [44]

    Yohan Chandrasukmana, Helena Margaretha, and Kie Van Ivanky Saputra. 2025. The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses.Examples and Counterexamples7 (2025), 100175

    work page 2025

  45. [45]

    Guy Chavent. 1979. Identification of distributed parameter systems: about the output least square method, its implementation, and identifiability. IFAC Proceedings Volumes12, 8 (1979), 85–97

    work page 1979

  46. [46]

    Haoxuan Chen, Yinuo Ren, Martin Renqiang Min, Lexing Ying, and Zachary Izzo. 2025. Solving inverse problems via diffusion-based priors: An approximation-free ensemble sampling approach.arXiv preprint arXiv:2506.03979(2025)

    work page arXiv 2025

  47. [47]

    Kangkang Chen, Xingjian Dong, Penglin Gao, Qian Chen, Zhike Peng, and Guang Meng. 2025. Physics-informed neural networks for topological metamaterial design and mechanical applications.International Journal of Mechanical Sciences301 (2025), 110489

    work page 2025

  48. [48]

    Ke Chen, Haizhao Yang, and Chugang Yi. 2026. Data Completion for Electrical Impedance Tomography by Conditional Diffusion Models.arXiv preprint arXiv:2602.07813(2026)

    work page arXiv 2026

  49. [49]

    Sirui Chen, Keenon Werling, Albert Wu, and C Karen Liu. 2022. Real-time model predictive control and system identification using differentiable simulation.IEEE Robotics and Automation Letters8, 1 (2022), 312–319

    work page 2022

  50. [50]

    Yuyao Chen, Lu Lu, George Em Karniadakis, and Luca Dal Negro. 2020. Physics-informed neural networks for inverse problems in nano-optics and metamaterials.Optics express28, 8 (2020), 11618–11633

    work page 2020

  51. [51]

    Yongbao Chen, Qiguo Yang, Zhe Chen, Chengchu Yan, Shu Zeng, and Mingkun Dai. 2023. Physics-informed neural networks for building thermal modeling and demand response control.Building and Environment234 (2023), 110149

    work page 2023

  52. [52]

    Ze Cheng, Zhuoyu Li, Wang Xiaoqiang, Jianing Huang, Zhizhou Zhang, Zhongkai Hao, and Hang Su. 2025. Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators. InInternational Conference on Machine Learning. PMLR, 10090–10105

    work page 2025

  53. [53]

    Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Marc Louis Klasky, and Jong Chul Ye. [n. d.]. Diffusion Posterior Sampling for General Noisy Inverse Problems. InThe Eleventh International Conference on Learning Representations

    work page

  54. [54]

    Hyungjin Chung, Suhyeon Lee, and Jong Chul Ye. 2024. Decomposed diffusion sampler for accelerating large-scale inverse problems. InInternational conference on learning representations, Vol. 2024. 38922–38949

    work page 2024

  55. [55]

    Hyungjin Chung, Dohoon Ryu, Michael T McCann, Marc L Klasky, and Jong Chul Ye. 2023. Solving 3d inverse problems using pre-trained 2d diffusion models. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition. 22542–22551

    work page 2023

  56. [56]

    Hyungjin Chung, Byeongsu Sim, Dohoon Ryu, and Jong Chul Ye. 2022. Improving diffusion models for inverse problems using manifold constraints. Advances in Neural Information Processing Systems35 (2022), 25683–25696

    work page 2022

  57. [57]

    Wai Tong Chung, Ki Sung Jung, Jacqueline H Chen, and Matthias Ihme. 2022. BLASTNet: A call for community-involved big data in combustion machine learning.Applications in Energy and Combustion Science12 (2022), 100087. Manuscript submitted to ACM 28 Zhentao Tan, Yuze Hao, Boyi Zou, Mingsheng Long, Yi Yang, and Gang Bao

    work page 2022

  58. [58]

    Shane Colburn and Arka Majumdar. 2021. Inverse design and flexible parameterization of meta-optics using algorithmic differentiation.Communi- cations Physics4, 1 (2021), 65

    work page 2021

  59. [59]

    Dario Coscia, Nicola Demo, and Gianluigi Rozza. 2024. Generative adversarial reduced order modelling.Scientific Reports14, 1 (2024), 3826

    work page 2024

  60. [60]

    2015.Applied computational aerodynamics: A modern engineering approach

    Russell M Cummings, William H Mason, Scott A Morton, and David R McDaniel. 2015.Applied computational aerodynamics: A modern engineering approach. Vol. 53. Cambridge University Press

    work page 2015

  61. [61]

    Giannis Daras, Hyungjin Chung, Chieh-Hsin Lai, Yuki Mitsufuji, Jong Chul Ye, Peyman Milanfar, Alexandros G Dimakis, and Mauricio Delbracio

    work page

  62. [62]

    A survey on diffusion models for inverse problems.arXiv preprint arXiv:2410.00083(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  63. [63]

    Maarten V de Hoop, Nikola B Kovachki, Matti Lassas, and Nicholas H Nelsen. 2025. Extension and neural operator approximation of the electrical impedance tomography inverse map.arXiv preprint arXiv:2511.20361(2025)

    work page arXiv 2025

  64. [64]

    Thomas Oliver de Jong, Khemraj Shukla, and Mircea Lazar. 2025. Deep operator neural network model predictive control.IEEE Open Journal of Control Systems(2025)

    work page 2025

  65. [65]

    Jonas Degrave, Federico Felici, Jonas Buchli, Michael Neunert, Brendan Tracey, Francesco Carpanese, Timo Ewalds, Roland Hafner, Abbas Abdolmaleki, Diego de Las Casas, et al. 2022. Magnetic control of tokamak plasmas through deep reinforcement learning.Nature602, 7897 (2022), 414–419

    work page 2022

  66. [66]

    Changyu Deng, Yizhou Wang, Can Qin, Yun Fu, and Wei Lu. 2022. Self-directed online machine learning for topology optimization.Nature communications13, 1 (2022), 388

    work page 2022

  67. [67]

    Yitong Deng, Hong-Xing Yu, Jiajun Wu, and Bo Zhu. [n. d.]. Learning Vortex Dynamics for Fluid Inference and Prediction. InThe Eleventh International Conference on Learning Representations

    work page

  68. [68]

    Alexander Denker, Fabio Margotti, Jianfeng Ning, Kim Knudsen, Derick Nganyu Tanyu, Bangti Jin, Andreas Hauptmann, and Peter Maass. 2025. Deep learning based reconstruction methods for electrical impedance tomography.arXiv preprint arXiv:2508.06281(2025)

    work page arXiv 2025

  69. [69]

    Thomas P Dussauge, Woong Je Sung, Olivia J Pinon Fischer, and Dimitri N Mavris. 2023. A reinforcement learning approach to airfoil shape optimization.Scientific Reports13, 1 (2023), 9753

    work page 2023

  70. [70]

    Weinan E and Bing Yu. 2018. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics6, 1 (Feb. 2018), 1–12

    work page 2018

  71. [71]

    Bradley Efron. 2011. Tweedie’s formula and selection bias.J. Amer. Statist. Assoc.106, 496 (2011), 1602–1614

    work page 2011

  72. [72]

    Mohamed Elrefaie, Angela Dai, and Faez Ahmed. 2024. Drivaernet: A parametric car dataset for data-driven aerodynamic design and graph-based drag prediction. InInternational Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 88360. American Society of Mechanical Engineers, V03AT03A019

    work page 2024

  73. [73]

    Mohamed Elrefaie, Florin Morar, Angela Dai, and Faez Ahmed. 2024. Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks.Advances in Neural Information Processing Systems37 (2024), 499–536

    work page 2024

  74. [74]

    David Erzmann and Soeren Dittmer. 2024. Equivariant neural operators for gradient-consistent topology optimization.Journal of Computational Design and Engineering11, 3 (2024), 91–100

    work page 2024

  75. [75]

    Haodong Feng, Haoren Zheng, Peiyan Hu, Hongyuan Liu, Chenglei Yu, Long Wei, Ruiqi Feng, Jinlong Duan, Dixia Fan, and Tailin Wu. 2025. FluidZero: Mastering Diverse Tasks in Fluid-Structure Interaction through a Single Generative Model. (2025)

    work page 2025

  76. [76]

    Mingquan Feng, Zhijie Chen, Yixin Huang, Yizhou Liu, and Junchi Yan. 2025. Optimal control operator perspective and a neural adaptive spectral method. InProceedings of the AAAI conference on artificial intelligence, Vol. 39. 14567–14575

    work page 2025

  77. [77]

    Marc Anton Finzi, Andres Potapczynski, Matthew Choptuik, and Andrew Gordon Wilson. 2023. A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks. InThe Eleventh International Conference on Learning Representations

    work page 2023

  78. [78]

    Bernat Font, Francisco Alcántara-Ávila, Jean Rabault, Ricardo Vinuesa, and Oriol Lehmkuhl. 2025. Deep reinforcement learning for active flow control in a turbulent separation bubble.Nature communications16, 1 (2025), 1422

    work page 2025

  79. [79]

    2014.Mathematical cardiac electrophysiology

    Piero Colli Franzone, Luca Franco Pavarino, and Simone Scacchi. 2014.Mathematical cardiac electrophysiology. Springer

    work page 2014

  80. [80]

    Claudia Giordana, Marina Mochi, and Francesco Zirilli. 1992. The numerical solution of an inverse problem for a class of one-dimensional diffusion equations with piecewise constant coefficients.SIAM J. Appl. Math.52, 2 (1992), 428–441

    work page 1992

Showing first 80 references.