Understanding the Difficulty of Solving Cauchy Problems with PINNs

arXiv:2405.02561v1 Announce Type: new
Abstract: Physics-Informed Neural Networks (PINNs) have gained popularity in scientific computing in recent years. However, they often fail to achieve the same level of accuracy as classical methods in solving differential equations. In this paper, we identify two sources of this issue in the case of Cauchy problems: the use of $L^2$ residuals as objective functions and the approximation gap of neural networks. We show that minimizing the sum of $L^2$ residual and initial condition error …

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