We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.

翻译：本文研究了使用带ReLU激活函数的物理信息神经网络（PINNs）求解多尺度输运方程的数值方法。为此，我们探讨了PINNs与最小二乘有限元法（LSFE）之间的类比关系，其共性在于两者都将偏微分方程求解重构为二次泛函的最小化问题。我们证明在扩散区域中，该方法无法收敛至正确极限，这与一阶LSFE的已知结论一致。通过引入扩散尺度变换可克服此问题，该结论与LSFE的理论结果完全吻合。我们提供了平板几何构型下的数值算例，验证了理论分析的正确性。