Theory Of Functional Connections Research Articles

Low-energy Earth–Moon transfers via Theory of Functional Connections and homotopy

Numerous missions leverage the weak stability boundary in the Earth–Moon–Sun system to achieve a safe and cost-effective access to the lunar environment. These transfers are envisaged to play a significant role in upcoming missions. This paper proposes a novel method to design low-energy transfers by combining the recent Theory of Functional Connections with a homotopic continuation approach. Planar patched transfer legs within the Earth–Moon and Sun–Earth systems are continued into higher-fidelity models. Eventually, the full Earth–Moon transfer is adjusted to conform to the dynamics of the planar Earth–Moon Sun-perturbed, bi-circular restricted four-body problem. The novelty lies in the avoidance of any propagation during the continuation process and final convergence. This formulation is beneficial when an extensive grid search is performed, automatically generating over 2000 low-energy transfers. Subsequently, these are optimized through a standard direct transcription and multiple shooting algorithm. This work illustrates that two-impulse low-energy transfers modeled in chaotic dynamic environments can be effectively formulated in Theory of Functional Connections, hence simplifying their overall design process. Moreover, its synergy with a homotopic continuation approach is demonstrated.

A Theory of Functional Connections-Based hp-Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems

This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians.

Tangential velocity constraint for orbital maneuvers with Theory of Functional Connections

Maneuvering a spacecraft in the cislunar space is a complex problem, since it is highly perturbed by the gravitational influence of both the Earth and the Moon, and possibly also the Sun. Trajectories minimizing the needed fuel are generally preferred in order to decrease the mass of the payload. A classical method to constrain maneuvers is mathematically modeling them using the Two Point Boundary Value Problem (TPBVP), defining spacecraft positions at the start and end of the trajectory. Solutions to this problem can then be obtained with optimization techniques like the nonlinear least squares conjugated with the Theory of Functional Connections (TFC) to embed the constraints, which recently became an effective method for deducing orbit transfers. In this paper, we propose a tangential velocity (TV) type of constraints to design orbital maneuvers. We show that the technique presented in this paper can be used to transfer a spacecraft (e.g. from the Earth to the Moon) and perform gravity assist maneuvers (e.g. a swing-by with the Moon). In comparison with the TPBVP, solving the TV constraints via TFC offers several advantages, leading to a significant reduction in computational time. Hence, it proves to be an efficient technique to design these maneuvers.

Theory of Functional Connections and Nelder–Mead optimization methods applied in satellite characterization

The growing population of man-made objects with the build up of mega-constellations and the multiplication of space debris not only increases the potential danger to all space vehicles and in-space infrastructures (including space observatories), but also poses a serious threat to astronomy and dark skies. Monitoring of this population requires precise satellite characterization, which is a challenging task that involves analyzing observational data such as position, velocity, and light curves using optimization methods. In this study, the applications of two optimization procedures are proposed and analyzed to determine the parameters associated with the dynamics of a satellite: one based on the Theory of Functional Connections (TFC) and another one based on the Nelder–Mead heuristic optimization algorithm. The TFC performs linear functional interpolation to embed the constraints of the problem into a functional. We propose to use this functional to analytically embed the observational data of a satellite into its equations of dynamics. After that, any solution will always satisfy the observational data. The second proposed procedure takes advantage of the Nelder–Mead algorithm, that does not require the gradient of the objective function, as alternative solution. The accuracy, efficiency, and dependency on the initial guess of each method is investigated, analyzed, and compared for several dynamical models. These methods can be used to obtain the physical parameters of a satellite from available observational data and for space debris characterization contributing to follow-up monitoring activities in space and astronomical observatories.

Form-finding of tensile membrane structures with strut and anchorage supports using physics-informed machine learning

Due to their aesthetic appeal, light weight, and capacity to span vast expanses of space, tensile membrane structures (TMS) are very popular these days. TMS are usually supported by frames and boundary edge cables, but sometimes also by struts and anchorage supports. Traditional mesh-based form-finding algorithms, typically developed for the first two boundary types, suffer from convergence issues stemming from the choice of initial shape, choice of pseudo material properties, mesh distortion, hyper-parameter selection, etc. In order to address the issues with conventional mesh-based form-finding frameworks as stated above, a mesh-free form-finding framework for TMS is proposed utilizing a gradient-enhanced physics-informed neural network. The theory of functional connections is employed to precisely meet the TMS boundary conditions without depending on an additional penalty function used in conventional physics-informed neural network frameworks, thereby eliminating errors related to boundary constraints. The challenge of incorporating compressive strut supports in TMS form-finding is addressed by employing the principle of inverted forces, and modelling them as tension elements. 24 numerical studies, with a diverse set of TMS shapes and boundary supports, including benchmark problems, are presented demonstrating the effectiveness and accuracy of the proposed method. This method also allows the designer to explore various parametric (architectural) designs of TMS.

A novel numerical approach for time-varying impulsive fractional differential equations using theory of functional connections and neural network

In this paper, we propose a physics-informed neural network-based scheme to solve time-varying impulsive fractional differential equations without any labeled data. At first, the existence and uniqueness of the solution of the impulsive fractional system are proved theoretically using the Bourdin state transition matrix-based solution representation and the Banach fixed point theorem. The proposed method uses the physics-informed neural network, extreme learning machines, and the theory of functional connections. A neural network is employed as the free function to construct the constrained expression. Based on the constrained expression, the linear system is trained using the extreme learning machine algorithm, and the nonlinear system is trained using the iterative least squares method, as the only trainable parameters are the output weights. The convergence and error analysis for the algorithm is derived. The neural network training is graphically demonstrated for the nonlinear system using the loss obtained in each iteration. For the linear case, the results are validated with the original solution, while for the nonlinear case, it is verified with the original solution and other classical methods. The primary use of the theory of functional connections is it helps the solution satisfy the physical constraint.

A novel method to approximate fractional differential equations based on the theory of functional connections

A novel method to approximate fractional differential equations based on the theory of functional connections

Physics-Informed Neural Networks for 2nd order ODEs with sharp gradients

In this work, four different methods based on Physics-Informed Neural Networks (PINNs) for solving Differential Equations (DE) are compared: Classic-PINN that makes use of Deep Neural Networks (DNNs) to approximate the DE solution;Deep-TFC improves the efficiency of classic-PINN by employing the constrained expression from the Theory of Functional Connections (TFC) so to analytically satisfy the DE constraints;PIELM that improves the accuracy of classic-PINN by employing a single-layer NN trained via Extreme Learning Machine (ELM) algorithm;X-TFC, which makes use of both constrained expression and ELM. The last has been recently introduced to solve challenging problems affected by discontinuity, learning solutions in cases where the other three methods fail. The four methods are compared by solving the boundary value problem arising from the 1D Steady-State Advection–Diffusion Equation for different values of the diffusion coefficient. The solutions of the DEs exhibit steep gradients as the value of the diffusion coefficient decreases, increasing the challenge of the problem.

Using the theory of functional connections to create periodic orbits with a linear variable thrust

This paper shows that the Theory of Functional Connections (TFC) can be used to find the thrust (given by a sub-optimal linear control law) required to create and maintain periodic orbits, like resonant orbits with the rotation of the Earth. Thus, the satellite will be located at the same position in the sky over the surface of the Earth every day, despite the perturbations. This facilitates the predictions of its motion and, thus, its tracking. The perturbations considered are: the J2 term of the Earth’s gravitational potential and the gravitational presence of the Sun and the Moon. The proposed approach is applied to two distinct periodic orbits. The first is an 8 h periodic orbit while the second is a 24 h geosynchronous periodic orbit. The fuel costs to maintain periodicity are evaluated and compared with other two methods: in the first method the perturbations are perfectly canceled by a variable thrust and in the second method a two-impulsive maneuver is applied as also obtained using TFC. Comparisons with the other two methods show that the one proposed in this paper can be used to lower the fuel costs.

Shaping low-thrust multi-target visit trajectories via theory of functional connections

This paper proposes a shaping method via theory of functional connections (TFC) to solve the coplanar low-thrust multi-target visit problems under the J2 perturbation. Three types of targets are considered including ground targets, space targets, and hybrid ground-space targets. The visit constraints for the ground and space targets under the linear J2-perturbed model are transformed into those under the two-body model with a correction term, which avoids directly solving the transfer trajectories with the J2 effect. Then, a unified shaping method via the TFC is proposed for the three types of targets. The multi-target visit problem is transformed into a nonlinear programming problem that analytically satisfies the initial conditions and the multi-target visit constraints. Several numerical examples show that the proposed method can realize rapid optimization for the whole transfer trajectories. Furthermore, the solved trajectories are successfully applied to the initial guess estimation of the Gauss pseudospectral method for further optimization.

Analysis of nonlinear Timoshenko–Ehrenfest beam problems with von Kármán nonlinearity using the Theory of Functional Connections

In this paper, the Theory of Functional Connections (TFC) is used to analyze static beams, accounting for the von Kármán nonlinearity and using the Timoshenko–Ehrenfest beam theory. The authors extend their earlier framework on linear beam bending problems to nonlinear bending problems using TFC. The TFC results and performance parameters are then compared to those of the Finite Element Method (FEM) to both validate the TFC solutions and compare computational efficiencies. In addition, this paper also focuses on the benefits of using TFC over FEM for stress analysis of beam bending problems. Also, a TFC methodology to solve buckling and free vibration problems for the linearized Timoshenko–Ehrenfest beam equations is introduced and validated. The results within this paper suggest that for most static beam bending problems, TFC provides more accurate solutions in terms of the residuals of the differential equations and a faster solution time when compared to the FEM using linear or quadratic approximations. Also, TFC has the added benefit of calculating stress fields that are continuous and smooth everywhere within the domain of the beam while FEM is limited to a piece-wise continuous stress field.

Time-energy optimal landing on planetary bodies via theory of functional connections

In this paper, we propose a unified approach to solve the time-energy optimal landing problem on planetary bodies (e.g. planets, moons, and asteroids). In particular, the indirect optimization method, based on the derivation of the first order necessary conditions from the Hamiltonian, is exploited and the Two-Point Boundary Value Problem arising from the application of the Pontryagin Minimum Principle is solved using the Theory of Functional Connections. The optimal landing trajectories are accurately computed with a computational time on the order of 10–100 ms, using a MATLAB implementation. The speed and accuracy of the proposed method makes it suitable for real time applications. The algorithm is applied and validated for the landing on large (Mars and Moon) and small (asteroids Gaspra and Bennu) planetary bodies.

Comparisons between the circular restricted three-body and bi-circular four body problems for transfers between the two smaller primaries

Important properties of the dynamics of a spacecraft can be obtained from the Circular Restricted Three Body Problem and the Bi-Circular Bi-planar Four Body Problem. In this work, both systems are compared under the perspective of the costs involved in a transfer between the smaller primaries. An analytical approach shows several properties of the perturbation due to the gravity of the Sun and the motion of the smaller primaries around it over a spacecraft in the region of interest, like its behavior at and around the barycenter or at any point in a circle around the Sun. The costs involved in transfers between the smaller primaries are numerically evaluated and analyzed using the newly developed Theory of Functional Connections. The results show that the influence of this perturbation over the costs is significant for systems like the Sun–Earth–Moon or Sun–Mars–Phobos. On the other hand, it is also shown that this influence may be negligible for other very different systems, like the Sun–Saturn–Titan or Sun–Ida–Dactyl. Maps of perturbation are drawn in the region of interest, which can be used for mission designers. Finally, a new approach to describe the influence of the Sun over the tides of the smaller primaries is proposed under the Four Body Problem model.

Theory of functional connections applied to quadratic and nonlinear programming under equality constraints

This paper introduces an efficient approach to solve quadratic and nonlinear programming problems subject to linear equality constraints via the theory of functional connections. This is done without using the traditional Lagrange multiplier technique. In particular, two distinct expressions (fully satisfying the equality constraints) are provided, to first solve the constrained quadratic programming problem as an unconstrained one for closed-form solution. Such expressions are derived by utilizing an optimization variable vector, which is called the free vector g by the theory of functional connections. In the spirit of this theory, for the equality constrained nonlinear programming problem, its solution is obtained by the Newton’s method combining with elimination scheme in optimization. Convergence analysis is supported by a numerical example for the proposed approach.

Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

This study shows how the Theory of Functional Connections (TFC) allows us to obtain fast and highly accurate solutions to linear ODEs involving integrals. Integrals can be constraints and/or terms of the differential equations (e.g., ordinary integro-differential equations). This study first summarizes TFC, a mathematical procedure to obtain constrained expressions. These are functionals representing all functions satisfying a set of linear constraints. These functionals contain a free function, g(x), representing the unknown function to optimize. Two numerical approaches are shown to numerically estimate g(x). The first models g(x) as a linear combination of a set of basis functions, such as Chebyshev or Legendre orthogonal polynomials, while the second models g(x) as a neural network. Meaningful problems are provided. In all numerical problems, the proposed method produces very fast and accurate solutions.

Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models

In this work, we apply a novel and accurate Physics-Informed Neural Network Theory of Functional Connections (PINN-TFC) based framework, called Extreme Theory of Functional Connections (X-TFC), for data-physics-driven parameters’ discovery of problems modeled via Ordinary Differential Equations (ODEs). The proposed method merges the standard PINNs with a functional interpolation technique named Theory of Functional Connections (TFC). In particular, this work focuses on the capability of X-TFC in solving inverse problems to estimate the parameters governing the epidemiological compartmental models via a deterministic approach. The epidemiological compartmental models treated in this work are Susceptible-Infectious-Recovered (SIR), Susceptible-Exposed-Infectious-Recovered (SEIR), and Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIRS). The results show the low computational times, the high accuracy, and effectiveness of the X-TFC method in performing data-driven parameters’ discovery systems modeled via parametric ODEs using unperturbed and perturbed data.

A TFC-based homotopy continuation algorithm with application to dynamics and control problems

A method for solving zero-finding problems is developed by tracking homotopy paths, which define connecting channels between an auxiliary problem and the objective problem. Current algorithms’ success highly relies on empirical knowledge, due to manually, inherently selected homotopy paths. This work introduces a homotopy method based on the Theory of Functional Connections (TFC). The TFC-based method implicitly defines infinite homotopy paths, from which the most promising ones are selected. A two-layer continuation algorithm is devised, where the first layer tracks the homotopy path by monotonously varying the continuation parameter, while the second layer recovers possible failures resorting to a TFC representation of the homotopy function. Compared to pseudo-arclength methods, the proposed TFC-based method retains the simplicity of direct continuation while allowing a flexible path switching. Numerical simulations illustrate the effectiveness of the presented method.

Analysis of Timoshenko–Ehrenfest beam problems using the Theory of Functional Connections

The numerical solution of boundary value problems in solid mechanics is dominated by the Finite Element Method (FEM). The present study provides an alternative numerical approach using the Theory of Functional Connections (TFC) for solving problems of this type. Here TFC is used to solve multiple beam problems using the Timoshenko–Ehrenfest beam theory which accounts for transverse shear strain. The results obtained are then compared to those obtained with both the FEM and the analytical approaches. For cases where the analytical solution is continuous and smooth, TFC out performs FEM both in terms of the L2 norm of the residuals and the solution time. When the analytical solution is continuous and smooth, TFC results in an L2 norm on the order of machine-error level precision. However, when the higher order derivatives of the analytical solution are discontinuous (i.e. when the applied load is discontinuous), TFC under performs compared to FEM. This is due to TFC approximating the solution by continuous functions when the true solution is piece-wise continuous. In terms of computational expense, TFC consistently solved the differential equations faster than FEM across each type of problem analyzed in this paper.

A Functional Interpolation Approach to Compute Periodic Orbits in the Circular-Restricted Three-Body Problem

In this paper, we develop a method to solve for periodic orbits, i.e., Lyapunov and Halo orbits, using a functional interpolation scheme called the Theory of Functional Connections (TFC). Using this technique, a periodic constraint is analytically embedded into the TFC constrained expression. By doing this, the system of differential equations governing the three-body problem is transformed into an unconstrained optimization problem where simple numerical schemes can be used to find a solution, e.g., nonlinear least-squares is used. This allows for a simpler numerical implementation with comparable accuracy and speed to the traditional differential corrector method.