Theory of Functional Connections

Theory of Functional Connections

The Theory of Functional Connections Project

 

The Theory of Functional Connections, developed by Daniele Mortari, is a mathematical framework designed to turn constrained problems into unconstrained problems. This is accomplished through the use of constrained expressions, which are functionals that represent the family of all possible functions that satisfy the problem’s constraints. The Theory of Functional Connections Project collects the methodologies and applications of the Theory of Functional Connections made by the efforts of researchers from several fields of study, such as Applied Mathematics, Optimal Control Theory, Space Engineering, Radiation, and Particle Transport.

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Collaborators

 

The University of Arizona (SSEL)

Texas A&M University

Politecnico di Milano

  • Francesco Topputo, Professor, Department of Aerospace Science & Technology.
  • Mauro Massari, Professor, Department of Aerospace Science & Technology.
  • Yang Wang, PhD Student, Department of Aerospace Science & Technology.

Sapienza University of Rome

 

Publications

  • Drozd, K., Furfaro, R., Schiassi, E., Johnston, H. and Mortari, D., 2021. Energy-optimal trajectory problems in relative motion solved via Theory of Functional Connections. Acta Astronautica, 182, pp.361-382. DOI: https://doi.org/10.1016/j.actaastro.2021.01.031
  • E. Schiassi, R. Furfaro, C. Leake, M. De Florio, H. Johnston, D. Mortari, Extreme Theory of Functional Connections: A Fast Physics-Informed Neural Network Method for Solving Ordinary and Partial Differential Equations, Neurocomputing (2021). DOI: https://doi.org/10.1016/j.neucom.2021.06.015
  • De Florio, M., Schiassi, E., Ganapol, B.D., Furfaro, R., "Physics-informed neural networks for rarefied-gas dynamics: Thermal creep flow in the Bhatnagar–Gross–Krook approximation", Physics of Fluids 33, 047110 (2021). DOI:  https://doi.org/10.1063/5.0046181
  • De Florio, M., Schiassi, E., Furfaro, R., Ganapol, B.D., Mostacci, D. (2020). Solutions of Chandrasekhar’s Basic Problem in Radiative Transfer via Theory of Functional Connections. Journal of Quantitative Spectroscopy & Radiative Transfer. p.107384. DOI: https://doi.org/10.1016/j.jqsrt.2020.107384
  • Schiassi, E., Leake, C., De Florio, M., Johnston, H., Furfaro, R., & Mortari, D. (2020). Extreme Theory of Functional Connections: A Physics-Informed Neural Network Method for Solving Parametric Differential Equations. arXiv :2005.10632. PDF
  • Leake, C. and Mortari, D., 2020. Deep theory of functional connections: A new method for estimating the solutions of partial differential equations. Machine learning and knowledge extraction, 2(1), pp.37-55. DOI: 10.3390/make2010004
  • Johnston, H., Leake, C. and Mortari, D., 2020. Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections. Mathematics, 8(3), p.397. DOI: 10.3390/math8030397
  • Johnston, H., Schiassi, E., Furfaro, R. and Mortari, D., 2020. Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via Theory of Functional Connections. arXiv preprint arXiv:2001.03572. PDF
  • Furfaro, R., & Mortari, D. 2020. Least-squares solution of a class of optimal space guidance problems via theory of connections. Acta Astronautica, 168, 92-103. DOI: https://doi.org/10.1016/j.actaastro.2019.05.050
  • Mortari, D. and Leake, C., 2019. The multivariate theory of connections. Mathematics, 7(3), p.296. DOI: 10.3390/math7030296
  • Wang, Y. and Topputo, F., 2019. Novel Homotopy Method via Theory of Functional Connections. arXiv preprint arXiv:1911.04899. PDF
  • Mai, T. and Mortari, D., 2019. Theory of functional connections applied to nonlinear programming under equality constraints. arXiv preprint arXiv:1910.04917. PDF
  • Leake, C., Johnston, H., Smith, L. and Mortari, D., 2019. Analytically embedding differential equation constraints into least squares support vector machines using the theory of functional connections. Machine learning and knowledge extraction, 1(4), pp.1058-1083. DOI: 10.3390/make1040060
  • Johnston, H., Leake, C., Efendiev, Y. and Mortari, D., 2019. Selected applications of the theory of connections: A technique for analytical constraint embedding. Mathematics, 7(6), p.537. DOI: 10.3390/math7060537
  • Drozd K., Furfaro R., & Mortari D. (2019). Constrained Energy-Optimal Guidance in Relative Motion via Theory of Functional Connections and Rapidly-Explored Random Trees. Conference: 2019 AAS/AIAA Astrodynamics Specialist Conference. Portland, ME, USA. PDF
  • Johnston, H. and Mortari, D., 2019. Least-squares solutions of boundary-value problems in hybrid systems. arXiv preprint arXiv:1911.04390. PDF
  • Mortari, D., Johnston, H. and Smith, L., 2019. High accuracy least-squares solutions of nonlinear differential equations. Journal of Computational and Applied Mathematics, 352, pp.293-307. DOI: 10.1016/j.cam.2018.12.007
  • Leake, C., 2018. Deep ToC: A new method for estimating the solutions of PDEs. arXiv preprint arXiv:1812.08625. PDF
  • Mortari, D., 2017. Least-squares solution of linear differential equations. Mathematics, 5(4), p.48. DOI: 10.3390/math5040048
  • Mortari, D., 2017. The theory of connections: Connecting points. Mathematics, 5(4), p.57. DOI: 10.3390/math5040057