Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

翻译：物理信息神经网络（PINNs）是一种用于求解偏微分方程（PDEs）的灵活深度学习框架，其建模范围涵盖从热传导到量子力学系统的各类现象。尽管具有灵活性，PINNs对其预测结果与真实解之间的偏差提供有限洞察，这阻碍了对其预测质量的信任。我们提出一种轻量级的事后处理方法，通过为PINN预测生成逐点误差估计来弥补这一不足。该方法为此类模型提供了一种自然的解释形式，不仅能识别预测是否正确，还能指出误差的位置和大小。对于线性偏微分方程，PINN近似解与真实解之间的误差满足与原问题相同的微分算子，但其源项由PINN的PDE残差驱动。我们使用有限差分方法对该误差方程进行数值求解，无需真实解的先验知识。在多个基准PDE上的评估表明，我们的方法能以较低计算成本生成精确的误差分布图，从而实现对PINNs的定向且可解释的验证。