SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 811-841, April 2024.
Abstract. Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.

中文翻译：

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wPINN：用于近似双曲守恒定律熵解的弱物理信息神经网络

SIAM 数值分析杂志，第 62 卷，第 2 期，第 811-841 页，2024 年 4 月。
摘要。物理信息神经网络 (PINN) 需要基础偏微分方程的解具有规律性，以保证精确的近似。因此，它们可能无法近似偏微分方程的不连续解，例如非线性双曲方程。为了改善这个问题，我们提出了一种新的 PINN 变体，称为弱 PINN (wPINN)，用于精确逼近标量守恒定律的熵解。wPINN 基于残差最小-最大优化问题的近似解（根据克鲁兹霍夫熵定义），以确定近似熵解以及测试函数的神经网络的参数。我们证明了 wPINN 所产生的误差的严格界限，并通过数值实验说明了它们的性能，以证明 wPINN 可以准确地近似熵解。