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<p>A frequent requirement in multidisciplinary optimization is determining one or more states that correspond to a feature of another state. For example, in order to perform structural analysis and optimization using a structural model, it might be necessary to provide the system loads at a time corresponding to a maximum displacement generated in a model of the system dynamics. Extracting these forces in a differentiable manner is essential for gradient-based optimization, but cannot be accomplished using an <Emphasis FontCategory="NonProportional">argmax</Emphasis> function. Attention mechanisms are a critical component of machine learning for natural language processing. Their mathematical form also provides a convenient method to pass information in a differentiable manner about one state that corresponds to important features of a second state. This approach has not been used, to the knowledge of the authors, in the context of multidisciplinary optimization. This paper describes the mathematical formulation of attention mechanisms to provide differentiability to state estimates passed from one model to another. Some important features and tunable parameters are discussed to improve the performance of the attention mechanism. Finally two examples, one simple and one an industrially relevant example of landing gear design, demonstrate the process.</p>

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Attention mechanisms for differentiable state approximation of dynamic systems in multidisciplinary optimization

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Andrew M. Ellis,
  • Craig A. Steeves

摘要

A frequent requirement in multidisciplinary optimization is determining one or more states that correspond to a feature of another state. For example, in order to perform structural analysis and optimization using a structural model, it might be necessary to provide the system loads at a time corresponding to a maximum displacement generated in a model of the system dynamics. Extracting these forces in a differentiable manner is essential for gradient-based optimization, but cannot be accomplished using an argmax function. Attention mechanisms are a critical component of machine learning for natural language processing. Their mathematical form also provides a convenient method to pass information in a differentiable manner about one state that corresponds to important features of a second state. This approach has not been used, to the knowledge of the authors, in the context of multidisciplinary optimization. This paper describes the mathematical formulation of attention mechanisms to provide differentiability to state estimates passed from one model to another. Some important features and tunable parameters are discussed to improve the performance of the attention mechanism. Finally two examples, one simple and one an industrially relevant example of landing gear design, demonstrate the process.