Physics-Informed Statistical Learning develops statistical learning methodologies that integrate complex data with scientific knowledge through a unified statistical learning framework. By embedding physical principles, differential equations, geometrical information and other domain-specific knowledge directly into statistical models, the resulting methods produce estimates that are not only statistically accurate, but also scientifically meaningful and physically consistent. Our research focuses on developing flexible methodologies for spatial, spatio-temporal and functional data observed over complex domains, including irregular planar domains, manifolds, volumetric domains and graph structures. It supports a broad range of statistical learning tasks, including regression, density and intensity estimation, quantile regression, functional principal component analysis, with statistical inference and uncertainty quantification. The methodology has applications across environmental sciences, climate science, medicine, neuroscience and engineering. An efficient open-source implementation of many of these methods is available through the fdaPDE library, a high-performance C++ library with an R interface and Python wrappers.

Below is a selection of representative publications illustrating the development of this research programme.

Download code:

   fdaPDE package on CRAN

   Development version on GitHub

 

  • Physics-informed data analysis of connectivity maps in grey matter volume

 

References

    1. Sangalli*, L.M., Ramsay, J.O., Ramsay, T.O. (2013),
      Spatial spline regression models,
      Journal of the Royal Statistical Society Ser. B, Statistical Methodology, 75, 4, 681-703. Spatial regresion • Irregular domains
      PDF     Postprint     Code
    2. Azzimonti#, L., Nobile, F., Sangalli*, L.M., Secchi, P. (2014),
      Mixed Finite Elements for spatial regression with PDE penalization,
      SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, No. 1, pp. 305-335. Physics-informed
      PDF     Postprint     Code
    3. Azzimonti#, L., Sangalli*, L.M., Secchi, P., Domanin, M., Nobile, F. (2015),
      Blood flow velocity field estimation via spatial regression with PDE penalization,
      Journal of the American Statistical Association, Theory and Methods, 110 (511), 1057-1071. Physics-informed
      PDF     Postprint     Supplementary material     Code
    4. Dassi, F., Ettinger#, B., Perotto, S., Sangalli, L.M. (2015),
      A mesh simplification strategy for a spatial regression analysis over the cortical surface of the brain,
      Applied Numerical Mathematics, Vol. 90, pp. 111-131. Manifold domains
      PDF     Postprint     Code
    5. Ettinger#, B., Perotto, S., Sangalli*, L.M. (2016),
      Spatial regression models over two-dimensional manifolds,
      Biometrika, 103 (1), 71-88. Manifold domains
      PDF     Postprint     Supplementary material     Code
    6. Lila#, E., Aston, J.A.D., Sangalli, L.M. (2016),
      Smooth Principal Component Analysis over two-dimensional manifolds with an application to Neuroimaging,
      Annals of Applied Statistics, 10 (4), 1854-1879. Functional PCA • Manifolds
      PDF     Postprint     Code
    7. Wilhelm#, M., Dede’, L., Sangalli, L.M., Wilhelm, P. (2016),
      IGS: an IsoGeometric approach for Smoothing on surfaces,
      Computer Methods in Applied Mechanics and Engineering, 302, 70-89. Manifold domains • Isogeometric analysis
      PDF     Postprint
    8. Wilhelm#, M., Sangalli*, L.M. (2016),
      Generalized Spatial Regression with Differential Regularization,
      Journal of Statistical Computation and Simulation, 86 (13), 2497-2518. Generalized linear models
      PDF     Postprint     Code
    9. Bernardi#, M.S., Sangalli*, L.M., Mazza#, G., Ramsay, J.O. (2017),
      A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province,
      Stochastic Environmental Research and Risk Assessment, 31 (1), 23-38. Space-time
      PDF     Postprint     Code
    10. Bernardi#, M.S., Carey, M., Ramsay, J.O., and Sangalli*, L.M. (2018),
      Modeling spatial anisotropy via regression with partial differential regularization,
      Journal of Multivariate Analysis, 167, 15-30. Anisotropy
      PDF     Postprint     Code
    11. Arnone#, E., Azzimonti#, A., Nobile, F., and Sangalli*, L.M. (2019),
      Modelling spatially dependent functional data via regression with differential regularization,
      Journal of Multivariate Analysis, 170, 275-295. Space-time • Physics-informed
      PDF     Postprint     Code
    12. Sangalli*, L.M. (2020),
      A novel approach to the analysis of spatial and functional data over complex domains,
      Quality Engineering, 32, 2, 181-190, Review
      followed by discussions and a rejoinder by the author.
      PDF     Postprint     Code
      Sangalli*, L.M. (2020),
      Rejoinder,
      Quality Engineering, 32, 2, 197-198.
      PDF
    13. Sangalli*, L.M. (2021),
      Spatial regression with partial differential equation regularization,
      International Statistical Review, 89 (3), 505–531. Review
      PDF     Code
    14. Ferraccioli#, F., Arnone#, E., Finos, L., Ramsay, J.O., Sangalli*, L.M. (2021),
      Nonparametric density estimation over complicated domains,
      Journal of the Royal Statistical Society Ser. B, Statistical Methodology, 83, 346–368. Density estimation
      PDF     Code
    15. Arnone#, E., Sangalli*, L.M., Vicini, A. (2022),
      Smoothing spatio-temporal data with complex missing data patterns,
      Statistical Modelling, DOI: 10.1177/1471082X211057959. Space-time • Missing data
      PDF     Postprint     Code
    16. Ponti#, L., Perotto, S., Sangalli, L.M. (2022),
      A PDE-regularized smoothing method for space-time data over manifolds with application to medical data,
      International Journal for Numerical Methods in Biomedical Engineering, 38 (12), e3650. Space-time • Physic-informed • Manifold domains
      PDF     Code
    17. Arnone#, E., Kneip, A., Nobile, F., Sangalli*, L.M. (2022),
      Some first results on the consistency of spatial regression with partial differential equation regularization,
      Statistica Sinica, 32, 209–238. Consistency
      PDF     Postprint
    18. Ferraccioli#, F., Sangalli*, L.M., Finos, L. (2022),
      Some first inferential tools for spatial regression with differential regularization,
      Journal of Multivariate Analysis, DOI: 10.1016/j.jmva.2021.104866. Inference
      PDF     Postprint
    19. Arnone#, E., Ferraccioli#, F., Pigolotti#, C., Sangalli*, L.M. (2022),
      A roughness penalty approach to estimate densities over two-dimensional manifolds,
      Computational Statistics and Data Analysis, 174, 107527. Density estimation • Manifold domains
      PDF     Postprint     Code
    20. Ferraccioli#, F., Sangalli, L.M., Finos, L. (2023),
      Nonparametric tests for semiparametric regression models,
      TEST, 32, 1106–1130. Inference
    21. Arnone#, E., Negri#, L., Panzica, F., Sangalli*, L.M. (2023),
      Analyzing data in complicated 3D domains: smoothing, semiparametric regression and functional principal component analysis,
      Biometrics, DOI: 10.1111/biom.13845. Volumes • Functional PCA
      PDF      Code
    22. Clementi#, L., Arnone, E., Santambrogio, M., Franceschetti, S., Panzica, F., Sangalli*, L.M. (2023),
      Anatomically compliant modes of variations: new tools for brain connectivity,
      PLOS ONE, DOI: 10.1371/journal.pone.0292450. Volumes • Functional PCA
      PDF      Code
    23. Arnone, E., De Falco, C., Formaggia, L., Meretti, G., Sangalli, L.M. (2023),
      Computationally efficient techniques for spatial regression with differential regularization,
      International Journal of Computer Mathematics, DOI: 10.1080/00207160.2023.2239944. Computation
      PDF      Code
    24. Arnone, E., Clemente#, A., Sangalli, L.M., Lila#, E., Ramsay, J., Formaggia, L. (2023),
      fdaPDE: Physics-Informed Spatial and Functional Data Analysis,
      R package available from CRAN. Software
      CRAN
    25. Begu#, B., Panzeri#, S., Arnone, E., Carey, M., Sangalli*, L.M. (2024),
      A nonparametric penalized likelihood approach to density estimation of space-time point patterns,
      Spatial Statistics, DOI: 10.1016/j.spasta.2024.100824. Point processes • Spatio-temporal
      PDF      Code
    26. Palummo#, A., Arnone, E., Formaggia, L., Sangalli*, L.M. (2024),
      Functional principal component analysis for incomplete space-time data,
      Environmental and Ecological Statistics, 31, 555–582. Functional PCA • Spatio-temporal • Gap-filling
      PDF      Code
    27. Castiglione#, C., Arnone, E., Bernardi, M., Farcomeni, A., Sangalli*, L.M. (2024),
      PDE-regularised spatial quantile regression,
      Journal of Multivariate Analysis, DOI: 10.1016/j.jmva.2024.105381. Quantile regression
      PDF      Code
    28. Tomasetto#, M., Arnone, E., Sangalli*, L.M. (2024),
      Modeling anisotropy and non-stationarity through physics-informed spatial regression,
      Environmetrics, DOI: 10.1002/env.2889. Physics-informed • Parameter cascading
      PDF      Code
    29. Cavazzutti#, M., Arnone, E., Ferraccioli, F., Galimberti#, C., Finos, L., Sangalli*, L.M. (2024),
      Sign-flip inference for spatial regression with differential regularisation,
      Stat, 13 (3), DOI: 10.1002/sta4.711. Inference
      PDF      Code
    30. Panzeri#, S., Clemente#, A., Arnone, E., Mateu, J., Sangalli*, L.M. (2025),
      Spatio-temporal intensity estimation for inhomogeneous Poisson point processes on linear networks: A roughness penalty method,
      Spatial Statistics, DOI: 10.1016/j.spasta.2025.100912. Point processes • Networks
      PDF      Code
    31. Di Battista#, I., De Sanctis#, M.F., Arnone, E., Castiglione, C., Palummo, A., Sangalli*, L.M. (2025),
      A semiparametric space-time quantile regression model,
      Journal of Nonparametric Statistics, DOI: 10.1080/10485252.2025.2593910. Quantile regression • Spatio-temporal
      PDF      Code
    32. De Sanctis#, M.F., Di Battista#, I., Arnone, E., Castiglione, C., Palummo, A., Bernardi, M., Ieva, F., Sangalli*, L.M. (2025),
      Exploring nitrogen dioxide spatial concentration via physics-informed multiple quantile regression,
      Environmental and Ecological Statistics, 32, 855–892. Multiple quantile regression
      PDF      Code
    33. Clemente#, A., Arnone, E., Mateu, J., Sangalli*, L.M. (2026),
      Nonparametric estimators over metric graphs,
      Biometrika, DOI: 10.1093/biomet/asag029. Networks • Density estimation
      PDF      Code
    34. De Sanctis#, M.F., Arnone, E., Ieva, F., Sangalli*, L.M. (2026),
      Modeling group heterogeneity in spatio-temporal data via physics-informed regression,
      Spatial Statistics, Vol 75, 101022. Mixed-effects • Spatio-temporal
      PDF      Code

    Other publications by L.M. Sangalli