Introduction: frontiers of applied inverse problems in science and engineering

Open AccessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations Share ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Smyl Danny, Hauptmann Andreas and Tallman Tyler 2025Introduction: frontiers of applied inverse problems in science and engineeringPhil. Trans. R. Soc. A.38320240056http://doi.org/10.1098/rsta.2024.0056SectionOpen AccessIntroductionIntroduction: frontiers of applied inverse problems in science and engineering Danny Smyl Danny Smyl https://orcid.org/0000-0002-6730-5277 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA [email protected] Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed Search for more papers by this author , Andreas Hauptmann Andreas Hauptmann Research Unit of Mathematical Sciences, University of Oulu, Oulu, Finland Department of Computer Science, University College London, London, UK Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed Search for more papers by this author and Tyler Tallman Tyler Tallman https://orcid.org/0000-0001-6572-4972 School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed Search for more papers by this author Danny Smyl Danny Smyl https://orcid.org/0000-0002-6730-5277 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA [email protected] Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed , Andreas Hauptmann Andreas Hauptmann Research Unit of Mathematical Sciences, University of Oulu, Oulu, Finland Department of Computer Science, University College London, London, UK Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed and Tyler Tallman Tyler Tallman https://orcid.org/0000-0001-6572-4972 School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA Contribution: Conceptualization, Writing – original draft Google Scholar Find this author on PubMed Published:25 September 2025https://doi.org/10.1098/rsta.2024.00561. IntroductionInverse problems have always been about peering through a mirror—looking backwards from the things we can measure to learn about the things we cannot directly see. In the last few years, that mirror has been polished by faster algorithms, smarter statistics and a new ally in machine learning. In the recent decades, it became apparent that inverse problems are not limited to the classical mathematical study of inverse boundary value problems, such as the Calderón problem [1], but have widespread practical applications. In fact, many scientific and engineering problems can be studied under the lens of inverse problems, revealing new viewpoints and enabling a new suite of methods [2,3]. Motivated by this insight, we edited this special issue on the frontiers of applied inverse problems in science and engineering, collecting thirteen papers, which were chosen to show how far the field has come and, more importantly, where it may be heading next. This special issue illustrates these advances by specifically highlighting how inverse problems are relevant and making an impact in a wide range of scientific fields, from aerospace and civil engineering to medical imaging, benefitting society and health at large. Each individual article tackles a practical question that used to feel out of reach and answers it with a blend of mathematical rigour and computational daring.Chung and co-authors show how to tame one of the biggest computational bottlenecks in Bayesian inference: picking the correct hyper-parameters. Their sample-average approximation turns what was once an art form into something approaching push-button automation, making fully probabilistic inversions feasible on everyday hardware. Zhang and colleagues push deeper into operator learning with their coefficient-to-basis network, a nimble architecture that changes its own resolution as it learns—a fine-tunability that could become a template for future solvers. Worden, Rogers and Preston return to the venerable Volterra series, refreshing it with modern identification tricks and reminding us that 'classical' does not have to mean 'dated'. Calvetti's team keeps physiology front and centre by applying separable hierarchical priors to extract and analyse muscle synergies in human locomotion, embedding physiological structure directly into their under-determined biomechanics models.If data are the new currency, material scientists are hitting the jackpot. Childs and collaborators use Bayesian machine learning to design ultra-high-performance concrete mixes that are known to be tedious to uncover by trial and error. Tian and co-authors' learning latent hardening framework folds micromechanical knowledge straight into a neural network, proving that black-box models can carry real metallurgical insight inside. Meanwhile, Esteghamati takes a similar philosophy into earthquake engineering, crafting an explainable surrogate that turns the once-esoteric loop of performance-based seismic design into an engineer-friendly inverse problem.Imaging, of course, is commonly considered the public face of inverse problems. Park, Jeon and Seo critically examine the mathematical foundations of low-dose, cost-effective dental cone beam computed tomograph—identifying research directions to enhance image quality by tackling metal-induced artefacts via alternative forward-model formulations beyond the Radon transform and by assessing the promise and limitations of deep learning-based methods. Ashraf and co-authors boost positron emission tomography reconstructions with an elegant deep-image prior, rescuing signal from the faintest of low-count acquisitions. Mueller demonstrates that electrical impedance tomography can pick up a newborn's ventilation and perfusion without radiating the infant—proof that gentle medicine and hard mathematics can coexist. Homa's group tackles the industrial end of the spectrum, segmenting eddy-current data with matching component analysis so that tiny defects in aerospace alloys no longer slip through the net. Nguyen and Le close the imaging loop with an orthogonality-sampling method for Maxwell's equations that works not just on simulations but on noisy experimental data.Finally, Karlsen and colleagues remind us that inverse problems can be explosive—literally. Their trilateration scheme infers both the yield and the point of origin of a blast wave from sparse pressure readings, offering a rapid forensic tool that could one day guide emergency response teams in real time.Taken together, the articles in this special issue deliver on our original intent: to illustrate how state-of-the-art mathematics, judicious statistics and physics-savvy machine learning are converging into a single theme. They lower computational barriers, widen the range of data we can trust and move several applications from 'promising' to 'practical'. Just as importantly, they flag the obstacles still ahead—from scaling to exascale hardware to keeping models honest when data are scarce. We hope the reader comes away energized, convinced that the next decade of inverse problems will be not only about seeing the unseen but about turning that vision into safer structures, stronger materials, cleaner images and, ultimately, better decisions.Data accessibilityThis article has no additional data.Declaration of AI useWe have not used AI-assisted technologies in creating this article.Authors' contributionsD.S.: conceptualization, writing—original draft; A.H.: conceptualization, writing—original draft; T.T.: conceptualization, writing—original draft.All authors gave final approval for publication and agreed to be held accountable for the work performed therein.Conflict of interest declarationThis theme issue was put together by the Guest Editor team under supervision from the journal's Editorial staff, following the Royal Society's ethical codes and best practice guidelines. The Guest Editor team invited contributions and handled the review process. Individual Guest Editors were not involved in assessing papers where they had a personal, professional or financial conflict of interest with the authors, or the research described. Independent reviewers assessed all papers. Invitation to contribute did not guarantee inclusion.FundingFunding was received from the Research Council of Finland (Flagship of Advanced Mathematics for Sensing Imaging and Modelling proj. 359186 and the Centre of Excellence of Inverse Modelling and Imaging proj. 353093, and the Academy Research Fellow proj. 338408).FootnotesOne contribution of 14 to a theme issue 'Frontiers of applied inverse problems in science and engineering'.© 2025 The Authors.Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Next Article VIEW FULL TEXTDOWNLOAD PDF FiguresRelatedReferencesDetails This Issue25 September 2025Volume 383Issue 2305Theme issue'Frontiers of applied inverse problems in science and engineering'compiled and edited by Danny Smyl, Andreas Hauptmann and Tyler Tallman Article InformationDOI:https://doi.org/10.1098/rsta.2024.0056PubMed:40994201Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Manuscript received25/06/2025Manuscript accepted25/06/2025Published online25/09/2025 License:© 2025 The Authors.Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Citations and impact Keywordsinverse problemsmachine learningcivil engineeringaerospace engineeringstructural engineering Subjectsstatistics