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<p>Simulation models are increasingly being built to predict and analyze nonlinear dynamic system behavior as a cheaper alternative to expensive physical testing. However, the simulation models may not perfectly represent the complicated physics due to the flawed understanding of the system, numerical approximations, and/or missing physical insight. The objective of this paper is to provide a better understanding of two nonlinear dynamic model bias correction strategies, namely <i>δ</i> learning (missing physics) and <i>δ</i> learning (machine learning (ML) prediction), which are two out of the six commonly used hybrid modeling strategies (also called physics-informed/enhanced machine learning methods). While these two <i>δ</i> learning strategies for model bias correction have been widely used in correcting static computational simulation models, their application to nonlinear dynamic simulation models is scarce. Even though there are a few applications of these two strategies to autonomous vehicle systems, battery state estimation, and river discharge prediction (for example), there are insufficient details provided about the theories and implementation details, which makes it difficult for practitioners to adopt these methods in practical applications. In this paper, we provide insights into the theories behind these two methods, explaining why they work, as well as details about the implementation procedures to facilitate the wide adoption. In addition, two examples, including a single-degree-of-freedom nonlinear oscillator and a six-story nonlinear shear-building model, are used to (1) demonstrate the effectiveness of these two hybrid modeling methods, and (2) comprehensively analyze the advantages and disadvantages of these two methods. Results show that both strategies can effectively correct a nonlinear dynamic simulation using a limited number of experiments. The <i>δ</i> learning (missing physics) method appears to be more accurate than the <i>δ</i> learning (ML prediction) method.</p>

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Towards a better understanding of model bias correction of nonlinear dynamic simulation models

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • Zhao Zhao,
  • Yichao Zeng,
  • Joshua W. Dyer,
  • Manuel A. Vega,
  • Michael D. Todd,
  • Zhen Hu

摘要

Simulation models are increasingly being built to predict and analyze nonlinear dynamic system behavior as a cheaper alternative to expensive physical testing. However, the simulation models may not perfectly represent the complicated physics due to the flawed understanding of the system, numerical approximations, and/or missing physical insight. The objective of this paper is to provide a better understanding of two nonlinear dynamic model bias correction strategies, namely δ learning (missing physics) and δ learning (machine learning (ML) prediction), which are two out of the six commonly used hybrid modeling strategies (also called physics-informed/enhanced machine learning methods). While these two δ learning strategies for model bias correction have been widely used in correcting static computational simulation models, their application to nonlinear dynamic simulation models is scarce. Even though there are a few applications of these two strategies to autonomous vehicle systems, battery state estimation, and river discharge prediction (for example), there are insufficient details provided about the theories and implementation details, which makes it difficult for practitioners to adopt these methods in practical applications. In this paper, we provide insights into the theories behind these two methods, explaining why they work, as well as details about the implementation procedures to facilitate the wide adoption. In addition, two examples, including a single-degree-of-freedom nonlinear oscillator and a six-story nonlinear shear-building model, are used to (1) demonstrate the effectiveness of these two hybrid modeling methods, and (2) comprehensively analyze the advantages and disadvantages of these two methods. Results show that both strategies can effectively correct a nonlinear dynamic simulation using a limited number of experiments. The δ learning (missing physics) method appears to be more accurate than the δ learning (ML prediction) method.