7 results
Search Results
2. Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution.
- Author
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Ma, Chuang, Huang, Jiangtao, Li, Daochun, Deng, Jun, Liu, Gang, Zhou, Lin, and Chen, Cheng
- Subjects
AERODYNAMIC load ,DRAG coefficient ,DRAG force ,AIRPLANE motors ,EQUATIONS - Abstract
This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Non-zero-sum differential games of delayed backward doubly stochastic systems and their application.
- Author
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Jie Xu, Rui Zhang, and Ruiqiang Lin
- Subjects
DIFFERENTIAL games ,STOCHASTIC systems ,STOCHASTIC differential equations ,NASH equilibrium ,DIFFERENTIAL equations ,DUALITY theory (Mathematics) ,MAXIMUM principles (Mathematics) - Abstract
In this paper, we investigate the non-zero-sum differential game problem based on the delayed backward doubly stochastic differential equation. We discuss the case where the controlled system contains time-delayed variables, and the time-delayed variables are different. We construct a new kind of adjoint equation consisting of a doubly stochastic differential equation and three simple differential equations. Under appropriate premises for such equations, we deduce a stochastic maximum principle as a necessary condition and a verification theorem for the Nash equilibrium point by using the duality method and convex variation technique. As an application, we apply our result to a linear delayed backward doubly stochastic non-zero-sum game problem to verify the effectiveness of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. A Functional Approach to Interpreting the Role of the Adjoint Equation in Machine Learning.
- Author
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Fekete, Imre, Molnár, András, and Simon, Péter L.
- Abstract
The connection between numerical methods for solving differential equations and machine learning has been revealed recently. Differential equations have been proposed as continuous analogues of deep neural networks, and then used in handling certain tasks, such as image recognition, where the training of a model includes learning the parameters of systems of ODEs from certain points along their trajectories. Treating this inverse problem of determining the parameters of a dynamical system that minimize the difference between data and trajectory by a gradient-based optimization method presents the solution of the adjoint equation as the continuous analogue of backpropagation that yields the appropriate gradients. The paper explores an abstract approach that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. The general maximum principle for discrete-time stochastic control problems.
- Author
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Song, Yuanzhuo and Wu, Zhen
- Subjects
- *
NOISE control , *CONVEX domains , *DISCRETE-time systems - Abstract
In this paper, we discuss the problem of discrete-time stochastic control with multiplicative noise, where the control domains may not be convex. We propose a novel approach inspired by the classical spike variation in continuous-time cases, but adapted to the discrete-time setting by perturbing in a random scale instead of time scale. Our approach allows us to obtain the maximum principle recursively without the need for variation equations or adjoint equations, which are typically required in previous methods. Moreover, we provide two maximum principles, one for the general case and the other when feedback forms exist. Our approach covers the previous results when the control domains are convex. Finally, we demonstrate the effectiveness of our new results through an example. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution
- Author
-
Chuang Ma, Jiangtao Huang, Daochun Li, Jun Deng, Gang Liu, Lin Zhou, and Cheng Chen
- Subjects
supersonic civil aircraft ,sonic boom ,adjoint equation ,optimization ,Motor vehicles. Aeronautics. Astronautics ,TL1-4050 - Abstract
This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force.
- Published
- 2024
- Full Text
- View/download PDF
7. Stochastic near-optimal control for a system with Markovian switching and Lévy noise.
- Author
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Kuang, Daipeng, Li, Jianli, Gao, Dongdong, and Luo, Danfeng
- Subjects
- *
ERROR functions , *HAMILTON'S principle function , *STOCHASTIC systems , *VARIATIONAL principles , *STOCHASTIC control theory , *INTEGRAL functions - Abstract
The near-optimal control conditions of a stochastic system are the main subject of this research work, which will extend some previous results. At first, the stochastic system and its adjoint equations are estimated based on certain mild assumptions on a convex control set. Additionally, utilizing various analysis techniques and Ekeland's variational principle, sufficient and necessary near-optimality conditions are obtained, independent of the second-order adjoint equation. Ultimately, an example of a biological, mathematical model is used to demonstrate the correctness of the theoretical analysis. • Based on some standard assumptions, the necessary as well as sufficient conditions of near optimality of stochastic system are given, which may be used to find the development of stochastic optimal control. • The systems can be seen as a generalization of [8–10, 21](e.g., the system degenerates to the model (1) in [8] when g = 0,n = 1.), and the results in [9,10,21] are included in our paper. • Especially, one of main results shows that any ϵ –optimal control nearly minimizes the Hamiltonian function in an integral form with an error order of "almost" ϵ 3 16 . • The another result shows that ϵ –minimum condition in terms of the Hamiltonian function in the integral form is sufficient for the near- optimality of order ϵ 1 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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