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An operator preconditioning perspective on training in physics-informed machine learning

by T. De Ryck and F. Bonnet and S. Mishra and E. de Bézenac

(Report number 2023-37)

Abstract
In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the \emph{Hermitian square} of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

Keywords: operator preconditioning, physics-informed neural networks, PINNs, scientific machine learning

@Techreport{DBMd23_1074,
  author = {T. De Ryck and F. Bonnet and S. Mishra and E. de Bézenac},
  title = {An operator preconditioning perspective on training in physics-informed machine learning},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-37.pdf },
  year = {2023}
}

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