Daniele Mortari

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Theory of Functional Connections

Books

  1. Mortari, D, Johnston, H, and Leake, C. “The Theory of Functional Connections,” Forthcoming book, 2021.

Thesis and dissertations

  1. De Florio, Mario, Accurate Solutions of the Radiative Transfer Problem via Theory of Connections, MS Thesis in Particles and Radiation Transport, Advisor D. Mostacci, Energy and Nuclear Engineering, University of Bologna (Italy), July 2019.
    ABSTRACT: In this thesis, a new approach to solving a class of radiative transfer problems is presented using the Theory of Functional Connections to solve the linear one-point boundary-value problem derived from the Boltzmann integrodifferential equation for the radiative transfer. The proposed algorithm resides in the category of numerical methods for the solution of transport equations and has been shown to be accurate and suitable for applications in atmospheric science and remote sensing.
  2. Johnston, Hunter, The Theory of Functional Connections: A Journey from Theory to Application, Ph.D. Dissertation, Advisor D. Mortari, Aerospace Engineering, Texas A&M University, Summer 2021.
    ABSTRACT: TFC is a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The functionals derived from this method, called \emph{constrained expressions}, analytically satisfy the imposed constraints and can be leveraged to transform constrained optimization problems to unconstrained ones. By simplifying the optimization problem, this technique has been shown to produce a numerical scheme that is faster, more accurate, and robust to poor initialization. The content of this dissertation details the complete development of the Theory of Functional Connections. First, the seminal paper on the Theory of Functional Connections is discussed and motivates the discovery of a more general formulation of the constrained expressions. Leveraging this formulation, a rigorous structure of the constrained expression is produced with associated mathematical definitions, theorems, and proofs. Furthermore, the second part of this dissertation explains how this technique can be used in the solution of ordinary differential equations providing a wide variety of examples with comparison to the state-of-the-art. The final part of this work focuses on unitizing the techniques and algorithms produced in the prior sections to explore the feasibility of using the Theory of Functional Connections to solve problems in real-time optimal control, namely optimal landing problems.
  3. Leake, Carl, The Multivariate Theory of Functional Connections: An n-Dimensional Constraint Embedding Technique Applied to Partial Differential Equations, Ph.D. Dissertation, Advisor D. Mortari, Aerospace Engineering, Texas A&M University, Summer 2021.
    ABSTRACT: This dissertation contains a comprehensive, self-contained presentation of the TFC theory beginning with simple univariate point constraints and ending with general linear constraints in $n$-dimensions; relevant mathematical theorems and clarifying examples are included throughout the presentation to expand and solidify the reader’s understanding. Furthermore, this dissertation describes how TFC can be applied to estimate the solutions of differential equations, its primary application to date. In addition, comparisons with other state-of-the-art algorithms that estimate the solutions of differential equations are included to showcase the advantages and disadvantages of the TFC approach. Lastly, the aforementioned concepts are leveraged to estimate solutions of differential equations from the field of flexible body dynamics.
  4. Yassopoulos, Christopher, A Comparison of the Finite Element Method and the Theory of Functional Connections with Regards to Solid Mechanics Problems in Engineering, MS Thesis, Advisor J.N. Reddy, Mechanical Engineering, Texas A&M University, Summer 2021.
    ABSTRACT: to be completed.
  5. Schiassi, Enrico, The Theory of Functional Connection and Physics-Informed Neural Networks for the Solution of Optimal Control Problems, Ph.D. Dissertation, Advisor R. Furfaro, Aerospace and Mechanical Engineering, University of Arizona, Fall 2021.
    ABSTRACT: to be completed.
  6. Gregory Arleth, Application of Theory of Functional Connections for Optimal Control of Nonlinear System, M.S. Thesis, Advisor J. Hurtado, Aerospace Engineering, Texas A&M University, May 2021.
    ABSTRACT: The scope of this research is to show that nonlinear, continuous, and computationally expensive equations can be simplified using ideas from the Theory of Functional Connections. Furthermore, we will show that the form these equations are simplified to are compatible with nonlinear optimal control problem algorithms such as Dynamic Programming. Practical examples involving hypersonic vehicles will also be used.

Journals

  1. Yassopoulos, C., Leake, C.,  Reddy, J.N., and Mortari, D. “Analysis of Timoshenko Beam Problems using the Theory of Functional Connections,” Special Issue “Computational Approaches to Mechanical Response Analysis of Structures at Diverse Scales,”‘ Journal Engineering Analysis with Boundary Elements, 2021, In print.
  2. Schiassi, E., Furfaro, R., Leake, C., De Florio, M., Johnston, H., and Mortari, D. “Extreme Theory of Functional Connections: A Fast Physics-Informed Neural Network Method for solving Ordinary and Partial Differential Equations,” Neurocomputing, 2021, In print.
  3. Johnston, H., Lo, W.M., and Mortari, D. “A Functional Interpolation Approach to Compute Periodic Orbits in the Circular Restricted Three-Body Problem”, Functional Interpolation (Mathematics), 2021, 9(11):1210. https://doi.org/10.3390/math9111210.
  4. Johnston, H. and Mortari, D. “Least-squares Solutions of Boundary-Value Problems in Hybrid Systems,” Journal of Computational and Applied Mathematics, 393, (2021) 113524.
  5. de Almeida, A., Johnston, H., Leake, C., and Mortari, D. “Fast 2-impulse non-Keplerian orbit-transfer using the Theory of Functional Connections,” The European Physical Journal Plus, 2021, in print.
  6. Drozd, K., Furfaro, R., Schiassi, E., Johnston, H., and Mortari, D. “Energy-Optimal Trajectory Problems in Relative Motion Solved via Theory of Functional Connections,” 2021, ACTA Astronautica. In print.
  7. Mortari, D. and Furfaro, R. “Univariate Theory of Functional Connections Applied to Component Constraints,” Mathematical and Computational Applications,” 2021,  26(1), 9.
  8. Mortari, D. and Arnas, D. “Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-rectangular 2-dimensional Domains,” Mathematics, 2020, 8(9), 1593.
  9. Furfaro, R. and Mortari, D. “Least-squares Solution of a Class of Optimal Guidance Problems via Theory of Connections,” ACTA Astronautica, March 2020, Vol. 168, pp.92-103.
  10. Leake, C. and Mortari, D. “Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations“, Machine Learning and Knowledge Extraction, 2020, Vol. 2, No. 1, pp. 37-55.
  11. Leake, C.,  Johnston, H., and Mortari, D. “The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations,” Mathematics, 2020, 8(8), 1303.
  12. Johnston, H., Leake, C., and Mortari, D. “Least-squares Solutions of Eighth-order Boundary Value Problems using the Theory of Functional Connections“, Mathematics, 2020, Vol. 8(3), No. 397.
  13. Johnston, H., Schiassi, E., Furfaro, R., and Mortari, D. “Fuel-Efficient Powered Descent Guidance on Planetary Bodies via the Theory of Functional Connections,” The Journal of Astronautical Sciences. 2020. DOI: 10.1007/s40295-020-00228-x
  14. Leake, C., Johnston, H., Smith, L., and Mortari, D. “Analytically Embedding Differential Equation Constraints into Least-Squares Support Vector Machines using the Theory of Functional Connections,” Machine Learning and Knowledge Extraction. 2019, 1(4), 1058-1083.
  15. Johnston, H., Leake, C., Efendiev, Y., and Mortari, D. “Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding,” Mathematics, 2019, 7(6), 537.
  16. Mortari, D. and Leake, C. “Multivariate Theory of Connections,” Mathematics, 2019, 7(3), 296.
  17. Mortari, D., Johnston, H., and Smith, L. “High Accurate Least-Squares Solutions of Nonlinear Differential Equations,” Journal of Computational and Applied Mathematics, 2019, Vol. 352, May 15, pp. 293-307.
  18. Mortari, D. “Least-Squares Solutions of Linear Differential Equations,” Mathematics, 2017, 5(4), 48. [2017 Best Paper Award, 2nd place]
  19. Mortari, D. “The Theory of Connections: Connecting Points,” Mathematics, 2017, 5(4), 57.

Conferences

  1. Mortari, D. “The Theory of Connections. Part 1: Connecting Points,” AAS 17-255, 2017 AAS/AIAA Space Flight Mechanics Meeting Conference, San Antonio, TX, February 5-9, 2017.
  2. Mortari, D. “Least-squares Solutions of Linear Differential Equations,” AAS 17-256, 2017 AAS/AIAA Space Flight Mechanics Meeting Conference, San Antonio, TX, February 5-9, 2017.
  3. Mortari, D. “The Theory of Connections with Application,” XVI Jornadas de Trabajo en Mecánica Celeste, Soria (Spain), June 19-21, 2017.
  4. Mortari, D., Johnston, H., and Smith, L. “Least-squares Solutions of Nonlinear Differential Equations,” Paper of the 2018 AAS/AIAA Space Flight Mechanics Meeting Conference, Kissimmee, FL, January 8-12, 2018.
  5. Mortari, D. and Furfaro, R. “Theory of Connections Applied to First-Order System of Ordinary Differential Equations Subject to Component Constraints,” AAS 18-230, 2018 AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19-23, 2018.
  6. Johnston, H. and Mortari, D. “Linear Differential Equations Subject to Relative, Integral, and Infinite Constraints,” AAS 18-273, 2018 AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19-23, 2018.
  7. Johnston, H. and Mortari, D. “Theory of Connections Solution to Perturbed Lambert’s Problem,” AAS 18-282, 2018 AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19-23, 2018.
  8. Furfaro, R. and Mortari, D. “Least-squares Solution of a Class of Optimal Guidance Problems,” AAS 18-362, 2018 AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19-23, 2018.
  9. Mortari, D. “The Theory of Functional Connections: Connecting Functions,” IAA-AAS-SciTech-072, Forum 2018, Peoples’ Friendship University of Russia, Moscow (Russia), November 13-15, 2018.
  10. Johnston, H. and Mortari, D. “Weighted Least-Squares Solutions of Over-Constrained Differential Equations,” IAA-AAS-SciTech-081, Forum 2018, Peoples’ Friendship University of Russia, Moscow (Russia), November 13-15, 2018.
  11. Mortari, D. “The Theory of Functional Connections: Current Status,” XIX Colóquio Brasileiro de Dinâmica Orbital, CBDO-2018. Instituto Nacional de Pesquisas Espaciais, São José dos Campos (Brasil), Dec. 3-7, 2018
  12. Drozd, K., Furfaro, R., and Mortari, D. “Constrained Energy-Optimal Guidance in Relative Motion via Theory of Functional Connections and Rapidly-Explored Random Trees,” AAS 19-662, 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  13. Mortari, D., Mai, T., and Efendiev, Y. “Theory of Functional Connections Applied to Nonlinear Programming under Equality Constraints,” AAS 19-675, of the 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  14. Schiassi, E., Furfaro, R., Johnston, H., and Mortari, D. “Fuel-efficient Powered Descent Guidance on Planetary Bodies via Theory of Functional Connection,” AAS 19-718, of the 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  15. Johnston, H., Leake, C., and Mortari, D. “An Analysis of the Theory of Functional Connections Subject to Inequality Constraints,” AAS 19-732, of the 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  16. Mortari, D. and Leake, C. “An Explanation and Implementation of the Multivariate Theory of Functional Connections via Examples,” AAS 19-734, 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  17. Johnston, H. and Mortari, D. “Orbit Propagation via the Theory of Functional Connections,” AAS 19-736, 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019.
  18. Mortari, D. “The Theory of Functional Connections: Current Status,” XXV International Congress of the Italian Association of Aeronautics and Astronautics, Rome (Italy), September 9-12, 2019.
  19. Furfaro, R., Drozd, K., and Mortari, D. “Energy-Optimal Rendezvous Spacecraft Guidance via Theory of Functional Connections,” 70th International Astronautical Congress 2019, IAF Astrodynamics Symposium, Washington, D.C., October 21-25, 2019.
  20. Schiassi, E., D’Ambrosio, A., Johnston, H., De Florio, M., Furfaro, R., Curti, F., and Mortari, D. “Physics-Informed Solution of Optimal Control Problems via Extreme Theory of Functional Connections”, AAS 20-524, Astrodynamics Specialist Conference, August 9-12, Lake Tahoe, CA.
  21. Schiassi, E., D’Ambrosio, A., Johnston, H., Furfaro, R., Curti, F., and Mortari, D. “Complete Energy Optimal Landing on Small and Large Planetary Bodies via Theory of Functional Connections”, AAS 20-557, Astrodynamics Specialist Conference, August 9-12, Lake Tahoe, CA.
  22. Furfaro, R., Schiassi, E., Drozd, K., and Mortari, D. “Physics-Informed Neural Networks and Theory of Functional Connections for Optimal Space Guidance Applications,” IAC 2020, 71-st International Astronautical Congress, 12-16 October 2020, Dubai, United Arab Emirates.