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<p>Physics-informed Machine Learning (PIML) methods are frameworks that integrate physical laws into machine learning models to solve forward and inverse problems involving differential and/or integral equations. These methods approximate solutions by minimizing the residuals of the governing equations and boundary conditions within the loss function of neural networks. Methodologies can be applied to solve inverse problems employing regularization techniques or calculating an adjoint equation in a variational approach. In the present study, Physics-informed approaches are evaluated, including the standard Physics-Informed Neural Networks (PINNs), Multiscale PINN and Kolmogorov–Arnold Networks (KANs). These three PIML formulations were tested to identify the initial condition in a heat conduction problem. These approaches can accurately reconstruct initial conditions without the need for explicit regularization and achieve high-resolution solutions, even in the presence of noisy data.</p>

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Solving backward heat transfer by physics-informed machine learning

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年6月3日

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  • Igor A. R. Oliveira,
  • Daniel A. S. Mendes,
  • Haroldo F. Campos Velho,
  • Eduardo S. Pereira

摘要

Physics-informed Machine Learning (PIML) methods are frameworks that integrate physical laws into machine learning models to solve forward and inverse problems involving differential and/or integral equations. These methods approximate solutions by minimizing the residuals of the governing equations and boundary conditions within the loss function of neural networks. Methodologies can be applied to solve inverse problems employing regularization techniques or calculating an adjoint equation in a variational approach. In the present study, Physics-informed approaches are evaluated, including the standard Physics-Informed Neural Networks (PINNs), Multiscale PINN and Kolmogorov–Arnold Networks (KANs). These three PIML formulations were tested to identify the initial condition in a heat conduction problem. These approaches can accurately reconstruct initial conditions without the need for explicit regularization and achieve high-resolution solutions, even in the presence of noisy data.