Solution of matrix Riccati differential equation for nonlinear singular system using neural networks

Samath, J. Abdul; Selvaraju, N.

doi:10.5120/575-181

Cited by 10 publications

(4 citation statements)

References 31 publications

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“…In order to solve this equation with a general initial or terminal condition, different fundamental solutions are proposed, including but not limited to Davison-Maki fundamental solution [35,89], symplectic fundamental solution [104], and min-plus fundamental solution [38,44,47,113]. Recently, there are some non-traditional methods developed for solving Riccati equations using ant colony programming [87], genetic programming [10] and neural networks [9,138].…”

Section: Implementations Of the Abstract Neural Network Architecturesmentioning

confidence: 99%

Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Darbon

Dower

Meng

2021

Preprint

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Solving high dimensional optimal control problems and corresponding Hamilton-Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which respectively represent the value function and the state feedback characterisation of the optimal control for certain class of high dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed using the deep neural network implementations of these abstract architectures. This work paves the way to leverage efficient dedicated hardware designed for neural networks to solve high dimensional optimal control problems and Hamilton-Jacobi PDEs.Keywords Optimal control • Hamilton-Jacobi partial differential equations • Neural networks • Grid-free numerical methods Research supported by AFOSR MURI FA9550-20-1-0358. Authors' names are given in last/family name alphabetical order.

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Section: Implementations Of the Abstract Neural Network Architecturesmentioning

confidence: 99%

Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Darbon

Dower

Meng

2021

Preprint

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“…-linear Riccati equations are also solved by using neural networks and the obtained solutions are compared by Runge-Kutta methods [36,37]. So in this paper, we solved the equations using 5th order Runge-Kutta method by choosing the initial condition as δ ℓ (0, k) = 0 and integrating to a large distance.…”

Section: Methodsmentioning

confidence: 99%

Constructing Inverse Scattering Potentials for α-α System using Physics Informed Machine Learning

sastri,

Sharma,

Awasthi

2023

Preprint

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An accurate way to incorporate long-range Coulomb interaction alongside short-range nuclear interaction remains challenging for theoretical physicists till date. Machine learning algorithms for global optimization, with physics equations guiding the process, are able to obtain robust models from available experimental data. In this article, we utilize the phase equation, which requires only potential and scattering energies as inputs, to obtain scattering phase shifts for the alpha-alpha system to construct the inverse potentials for its S, D, and G states. By incorporating the phase equation within an iterative loop of optimization code, mean absolute percentage error(MAPE) is minimized to obtain the best model parameters for a chosen reference potential. The key to successfully incorporating Coulomb interaction is in designing a reference potential. Here, a combination of two smoothly joined Morse functions, one regular followed by an inverted one, is considered. While the former takes care of short-range nuclear and Coulomb interactions, the latter accounts for expected barrier height due to the long-range Coulomb part that dominates once nuclear interaction subsides. The constructed inverse potentials for S, D, and G states have resulted in MAPE of 0.9, 0.5, and 0.4 and resonances(experimental) at 0.1240(0.0918), 2.95(3.03), and 11.89(11.35) respectively.

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“…In other words, the authors used the residual of the MDRE as the loss function. Following the work of Reference [16], more complex MDREs from singular, stochastic, and fuzzy systems were quickly solved [17][18][19][20][21]. Each neural network in these studies was trained with the Levenberg-Marquardt algorithm and validated by comparisons with the Runge-Kutta algorithm.…”

Section: Related Workmentioning

confidence: 99%

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

Drozd,

Furfaro,

Schiassi

et al. 2023

Mathematics

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In this manuscript, we explore how the solution of the matrix differential Riccati equation (MDRE) can be computed with the Extreme Theory of Functional Connections (X-TFC). X-TFC is a physics-informed neural network that uses functional interpolation to analytically satisfy linear constraints, such as the MDRE’s terminal constraint. We utilize two approaches for solving the MDRE with X-TFC: direct and indirect implementation. The first approach involves solving the MDRE directly with X-TFC, where the matrix equations are vectorized to form a system of first order differential equations and solved with iterative least squares. In the latter approach, the MDRE is first transformed into a matrix differential Lyapunov equation (MDLE) based on the anti-stabilizing solution of the algebraic Riccati equation. The MDLE is easier to solve with X-TFC because it is linear, while the MDRE is nonlinear. Furthermore, the MDLE solution can easily be transformed back into the MDRE solution. Both approaches are validated by solving a fluid catalytic reactor problem and comparing the results with several state-of-the-art methods. Our work demonstrates that the first approach should be performed if a highly accurate solution is desired, while the second approach should be used if a quicker computation time is needed.

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Solution of matrix Riccati differential equation for nonlinear singular system using neural networks

Cited by 10 publications

References 31 publications

Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Constructing Inverse Scattering Potentials for α-α System using Physics Informed Machine Learning

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

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