We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference “ground truth”, and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.

中文翻译：

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用于解决黎曼问题的梯度湮没 PINN：在相对论流体动力学中的应用

我们提出了一种基于物理信息神经网络 (PINN) 的新颖方法，用于求解允许不连续解的偏微分方程组。我们的方法称为梯度消灭 PINN (GA-PINN)，引入了一种修改后的损失函数，迫使模型部分忽略物理变量中的高梯度，这是通过引入合适的加权函数来实现的。该方法依赖于一组超参数来控制物理损失中梯度的处理方式。我们的方法的性能通过解决特殊相对论流体动力学中的黎曼问题得到证明，并在经典欧拉方程的背景下扩展了 PINN 的早期研究。使用 GA-PINN 模型获得的解正确地描述了不连续性的传播速度并敏锐地捕获了相关的跳跃。我们使用相对误差将我们的结果与用作参考“基本事实”的特殊相对论黎曼问题的精确解以及通过二阶中心冲击捕获方案获得的相应误差进行比较。在研究的所有问题中，GA-PINN 模型达到的精度与冲击捕获方案获得的精度相当，总体上实现了优于基线 PINN 算法的性能。值得强调的另一个好处是，我们基于 PINN 的方法避免了从守恒变量的状态向量中恢复原始变量的成本高昂，这是相对论流体动力学方程的基于网格的解决方案的众所周知的缺点。由于其固有的通用性和处理陡峭梯度的能力，本文讨论的 GA-PINN 方法可能成为模拟天体物理学和粒子物理学中相对论流的宝贵工具，其特点是普遍存在不连续解。