Physics-Informed Extreme Theory of Functional Connections Applied to Data-Driven Parameters Discovery of Epidemiological Compartmental Models

In this work we apply a novel, accurate, fast, and robust physics-informed neural network framework for data-driven parameters discovery of problems modeled via parametric ordinary differential equations (ODEs) called the Extreme Theory of Functional Connections (X-TFC). The proposed method merges two recently developed frameworks for solving problems involving parametric DEs, 1) the Theory of Functional Connections (TFC) and 2) the Physics-Informed Neural Networks (PINN). 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 (SEIR). The results show the low computational times, the high accuracy and effectiveness of the X-TFC method in performing data-driven parameters discovery of systems modeled via parametric ODEs using unperturbed and perturbed data

Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via Theory of Functional Connections

In this paper we present a new approach to solve the fuel-efficient powered descent guidance problem on large planetary bodies with no atmosphere (e.g. the Moon or Mars) using the recently developed Theory of Functional Connections. The problem is formulated using the indirect method which casts the optimal guidance problem as a system of nonlinear two-point boundary value problems. Using the Theory of Functional Connections, the problem constraints are analytically embedded into a "constrained expression," which maintains a free-function that is expanded using orthogonal polynomials with unknown coefficients. The constraints are satisfied regardless of the values of the unknown coefficients which convert the two-point boundary value problem into an unconstrained optimization problem. This process casts the solution into the admissible subspace of the problem and therefore simple numerical techniques can be used (i.e. in this paper a nonlinear least-squares method is used). In addition to the derivation of this technique, the method is validated in two scenarios and the results are compared to those obtained by the general purpose optimal control software, GPOPS-II. In general, the proposed technique produces solutions of $\mathcal{O}(10^{-10})$. Additionally, for the proposed test cases, it is reported that each individual TFC-based inner-loop iteration converges within 6 iterations, each iteration exhibiting a computational time between 72 and 81 milliseconds within the MATLAB legacy implementation. Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time.Comment: 17 pages, 10 figures, 6 table

Theoretical Evaluation of Anisotropic Reflectance Correction Approaches for Addressing Multi-Scale Topographic Effects on the Radiation-Transfer Cascade in Mountain Environments

Research involving anisotropic-reflectance correction (ARC) of multispectral imagery to account for topographic effects has been ongoing for approximately 40 years. A large body of research has focused on evaluating empirical ARC methods, resulting in inconsistent results. Consequently, our research objective was to evaluate commonly used ARC methods using first-order radiation-transfer modeling to simulate ASTER multispectral imagery over Nanga Parbat, Himalaya. Specifically, we accounted for orbital dynamics, atmospheric absorption and scattering, direct- and diffuse-skylight irradiance, land cover structure, and surface biophysical variations to evaluate their effectiveness in reducing multi-scale topographic effects. Our results clearly reveal that the empirical methods we evaluated could not reasonably account for multi-scale topographic effects at Nanga Parbat. The magnitude of reflectance and the correlation structure of biophysical properties were not preserved in the topographically-corrected multispectral imagery. The CCOR and SCS+C methods were able to remove topographic effects, given the Lambertian assumption, although atmospheric correction was required, and we did not account for other primary and secondary topographic effects that are thought to significantly influence spectral variation in imagery acquired over mountains. Evaluation of structural-similarity index images revealed spatially variable results that are wavelength dependent. Collectively, our simulation and evaluation procedures strongly suggest that empirical ARC methods have significant limitations for addressing anisotropic reflectance caused by multi-scale topographic effects. Results indicate that atmospheric correction is essential, and most methods failed to adequately produce the appropriate magnitude and spatial variation of surface reflectance in corrected imagery. Results were also wavelength dependent, as topographic effects influence radiation-transfer components differently in different regions of the electromagnetic spectrum. Our results explain inconsistencies described in the literature, and indicate that numerical modeling efforts are required to better account for multi-scale topographic effects in various radiation-transfer components.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Extreme Theory of Functional Connections for Problems Involving Differential Equations with Applications to Optimal Control Problems

In recent years, Neural Networks (NNs) have become widespread across many scientific fields. This requires designing them to satisfy application-specific constraints, such as conservation laws, symmetries, or other domain-specific knowledge. These constraints are usually imposed as soft penalties during network training and act as regularizers within the loss function. Physics-Informed Neural Networks (PINNs) are examples of this philosophy, where the outputs of the network are constrained to approximately satisfy specific physics laws, modeled as a set of Differential Equations (DEs).This dissertation presents a novel, accurate, fast, flexible, reliable, and robust PINN framework for forward and inverse problems governed by DEs. This framework is called Extreme Theory of Functional Connections (X-TFC). The proposed method, for the first time, merges the Theory of Functional Connections (TFC) and the Extreme Learning Machines (ELM). TFC enables functional interpolation for a large class of mathematical objects. According to TFC, any mathematical problem solution can be represented via the Constrained Expressions (CEs). The CEs are functional. These functionals are the sum of a free function and a functional that analytically satisfies the problem constraints. When applied to problems involving DEs, the mathematical problem is represented by the DE itself, where the constraints are the DE initial and/or boundary conditions. The DE solution is approximated with the CE, where the free function, according to X-TFC, is chosen to be a shallow NN trained via the Extreme Learning Machine (ELM) algorithm. ELM algorithm randomly selects input weights and biases, which are not tuned during the training, leaving the output weights as the only trainable parameters. Thus, the training is performed with a fast and robust least-squares. This work shows the X-TFC advantage over the classic PINN framework and some of its variants. In particular, it will show many X-TFC applications in solving forward and inverse problems involving DEs. X-TFC has been applied to physics-driven solutions of nuclear reactor dynamics (e.g., point kinetic equations with temperature feedback). Another application is for physics-driven solutions of Optimal Control Problems (OCPs), both via the application of the indirect method (e.g., Pontryagin Minimum Principle) and the Bellman Principle of Optimality (BPO), both for general and aerospace OCPs. X-TFC has also been applied for data-physics-driven parameter discovery of epidemiological models that regulate virus spread.Dissertation not available (per author's request

Bayesian inversion of coupled radiative and heat transfer models for asteroid regoliths and lakes

estimation, the quantities we seek to retrieve are considered as random variables. The randomness includes the uncertainty regarding their true values.Weintend to use this approach to perform inversion of coupled radiative and heat transfer models for asteroid regoliths and lakes. The Bayesian inversion of this kind of models allows estimating optical and thermodynamic properties of the systems considered, and also allows finding any correlation among these properties; that would be quite difficult to find with the classical approaches

Physics-informed Neural Networks for Optimal Intercept Problem,

The novel Extreme Theory of Functional Connections (X-TFC) method is employed to solve the optimal intercept problem. With X-TFC, for the first time, Theory of Functional Connections (TFC) and shallow Neural Networks (NNs) trained via the Extreme Learning Machine (ELM) algorithm are brought together as a class of PINN methods and applied to solving a broad class of ODEs and PDEs. In particular, the unknown solutions (in strong sense) of the ODEs and PDEs are approximated via particular expressions, called constrained expression (CEs), defined within TFC. A CE is a functional that always analytically satisfies the specified constraints and has a free-function that does not affect the specified constraints. In the X-TFC method, the free-function is a single-layer NN, trained via ELM algorithm. According to the ELM algorithm, the unknown constant coefficients appear linearly and thus, a least-squares method (for linear problems) or an iterative least-square method (for nonlinear problems) is used to compute the unknowns by minimizing the residual of the differential equations. In this work, the differential equations are represented by the system arising from the indirect method formulation of optimal control problems, which exploits the Hamiltonian function and the Pontryagin Maximum/Minimum Principle (PMP) to obtain a Two-Point Boundary Value Problem. The proposed method is tested by solving the Feldbaum problem and the minimum time-energy optimal intercept problem. It is shown that the major advantage of this method is the comparable accuracy with respect to the state of the art methods for the solution of optimal control problems along with an extremely fast computational time. In particular, the low computational time makes the proposed method suitable for real-time applications

PHYSICS-INFORMED NEURAL NETWORKS APPLIED TO A SERIES OF CONSTRAINED SPACE GUIDANCE PROBLEMS

The newly developed method named Extreme Theory of Functional Connections, or X-TFC, is exploited in this paper to solve constrained optimal control problems. This framework belongs to the family of Physics-Informed Neural Networks (PINNs), and it exploits the so-called Constrained Expressions (CEs) to approximate the latent (unknown) solutions. These expressions, developed within the Theory of Functional Connections (TFC) framework, are the sum of a freechosen function, and a functional that always analytically satisfies the boundary conditions. According to the X-TFC method, the free function is a single layer neural network trained via Extreme Learning Machine (ELM) algorithm. Optimal control problems are treated via indirect method, based on the Hamiltonian of the problem and the Pontryagin Minimum Principle to obtain the optimal control and the first order necessary conditions. Within this formulation, inequality constraints are considered by introducing new variables and additional terms in the cost function, and in the Hamiltonian. Moreover, saturation functions are used to consider the boundaries of inequality constraints. X-TFC is then employed to solve the boundary value problem that arises from the indirect method. Since the boundary conditions are a priori satisfied, accurate results are obtained with a low computational time

PHYSICS-INFORMED NEURAL NETWORKS FOR OPTIMAL PROXIMITY MANEUVERS WITH COLLISION AVOIDANCE AROUND ASTEROIDS

Exploring small planetary bodies, such as asteroids, is essential in understanding our planetary evolution and formation. For this reason, space agencies design space missions to explore these bodies. Thus, it is necessary to develop tools to compute optimal proximity maneuvers and trajectories around asteroids accurately. However, one of the main difficulties when dealing with asteroids is their irregular shapes, which can eventually lead a spacecraft to unexpected impacts if its trajectory is not designed carefully. To this end, this paper shows how it is possible to design optimal trajectories with collision avoidance around asteroids so that the spacecraft can avoid impacts with the irregular shape of an asteroid. We do so by employing the Rapidly-Explored Random Tree (RRT*) technique, which allows us to connect multiple arches of trajectory to avoid obstacles. In particular, every single optimal arch is computed via the indirect method exploiting Physics-Informed Neural Networks (PINNs). This is done by learning the Two-Point Boundary Value Problem (TPBVP) solution arising from applying the Pontryagin Minimum Principle (PMP) to the optimal control problem. The Extreme Theory of Functional Connections (X-TFC) is employed among the PINN frameworks because it analytically satisfies the boundary constraints. The proposed method is tested to design optimal trajectories around asteroids Gaspra and Bennu while avoiding impacts with their surfaces

Energy-optimal trajectory problems in relative motion solved via Theory of Functional Connections

In this paper, we present a new approach for solving a broad class of energy-optimal trajectory problems in relative motion using the recently developed Theory of Functional Connections (TFC). A total of four problem cases are considered and solved, i.e. rendezvous and intercept with fixed and free final time. Each problem is constrained and formulated using an indirect approach which casts the optimal trajectory problem as a system of linear or nonlinear two-point boundary value problems for the fixed and free final time cases, respectively. Using TFC, we convert each two-point boundary value problem into an unconstrained problem by analytically embedding the boundary constraints into a “constrained expression.” The latter includes a free-function that is expanded using Chebyshev polynomials with unknown coefficients. Regardless of the values of the unknown coefficients, the boundary constraints are satisfied and simple optimization schemes can be employed to numerically solve the problem (e.g. linear and nonlinear least-square methods). To validate the proposed approach, the TFC solutions are compared with solutions obtained via an analytical based method as well as direct and indirect numerical methods. In general, the proposed technique produces solutions to machine level accuracy. Additionally, for the cases tested, it is reported that computational run-time within the MATLAB implementation is lower than 28 and 300 ms for the fixed and free final time problems respectively. Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time.24 month embargo; first published online 4 February 2021This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]