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Parametric optimal control problems governed by partial differential equations (PDEs) are widely found in scientific and engineering applications. Traditional grid-based numerical methods for such problems generally require repeated solutions of PDEs with different parameter settings, which is computationally prohibitive especially for problems with high-dimensional parameter spaces. Although recently proposed neural network methods make it possible to obtain the optimal solutions simultaneously for different parameters, challenges still remain when dealing with problems with complex constraints. In this paper, we propose AONN, an adjoint-oriented neural network method, to overcome the limitations of existing approaches in solving parametric optimal control problems. In AONN, the neural networks are served as parametric surrogate models for the control, adjoint and state functions to get the optimal solutions all at once. In order to reduce the training difficulty and handle complex constraints, we introduce an iterative training framework inspired by the classical direct-adjoint looping (DAL) method so that penalty terms arising from the Karush-Kuhn-Tucker (KKT) system can be avoided. Once the training is done, parameter-specific optimal solutions can be quickly computed through the forward propagation of the neural networks, which may be further used for analyzing the parametric properties of the optimal solutions. The validity and efficiency of AONN is demonstrated through a series of numerical experiments with problems involving various types of parameters.

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In this work, novel plasticizing biodegradable poly (lactic acid) (PLA) composites were prepared by melt blending of jute and tung oil anhydride (TOA), and the physical and mechanical properties of PLA/jute/TOA composites were tested and characterized. The impact strength of PLA/jute/TOA composites significantly increases with increasing the content of TOA. The SEM images of fracture surface of PLA/jute/TOA composites become rough after the incorporation of TOA. In addition, TOA changes the crystallization temperature and decomposition process of PLA/jute/TOA composites. With increasing the amount of TOA, the value of storage modulus (E′) of PLA/jute/TOA composites gradually increases. The complex viscosity (η*) values for all samples reduce obviously with increasing the frequency, which means that the pure PLA and PLA/jute/TOA composites is typical pseudoplastic fluid. This is attributed to the formation of crosslinking, which restricts the deformation of the composites.

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