AI 中文总结
本文指出PINNs中残差更低的解未必更优，在多数匹配求解器对中残差与结构恒等式的排序会反转，提出含结构恒等式等三项的可容许性门解决该问题，还发现预设结构先验的97%结构与物理信息无关。
AI 中文摘要
物理信息神经网络（PINNs）通常通过单一的聚合残差进行评估，假设残差越小则解越优。本文针对三个带约束的偏微分方程（PDE）系统直接验证该假设，发现其会系统性失效。在仅是否将核心结构恒等式硬编码或惩罚化的匹配求解器对中，惩罚化变体常获得更低的方程残差，却在该恒等式上的违反程度达数个数量级，导致精确变体被错误地评为更差。在涵盖两个系统、四种网络变体、八个随机种子的64对匹配求解器中，这种反向现象发生在83%的案例中（95%威尔逊置信区间：72%至90%），各系统的发生率为72%至94%。针对六种架构——多层感知器（MLP）、约束物理信息神经网络（cPINN）、跨域物理信息神经网络（XPINN）、hp-VPINN、物理信息深度算子网络（DeepONet）以及傅里叶神经算子（FNO）——的测试显示，48对中有46对的排名发生反转，表明这种变异性取决于问题本身，而非特定的近似器。第三个更大的涡量-流函数问题也呈现相同的排序：残差最优的求解器在结构恒等式上的违反程度超出容差6个数量级以上，尽管残差仅存在9.37%的优势。由于标量损失无法暴露该问题，本文引入一种按词典顺序排列的可容许性门，覆盖结构恒等式、边界迹线和必须独立于方程残差消失的可解性积分，所有三项均通过后残差方可竞争。该门可捕捉三类伪影，但遗漏第四类：预设结构先验生成的场通过所有单次运行检查，然而删除源项后发现，报告结构的97%在物理信息被移除后仍留存。论文附带参考数据和图表。
英文摘要
Physics-informed neural networks (PINNs) are commonly evaluated via a single aggregate residual, assuming a smaller residual indicates a better solution. Testing this directly across three constrained PDE systems, I find this assumption can systematically fail. In matched pairs of solvers differing only in whether a defining structural identity is hard-wired or penalized, the penalized variant frequently attains a lower equation residual while violating that identity by several orders of magnitude, causing the exact variant to be falsely ranked worse. Over 64 matched pairs spanning two systems, four network variants, and eight seeds, this inversion occurs in 83\% of cases (95\% Wilson CI: 72--90\%), with rates from 72\% to 94\% across systems. Testing across six architectures--MLP, cPINN, XPINN, hp-VPINN, and physics-informed DeepONet and FNO--inverts the ranking in 46 of 48 pairs, indicating that this variability is problem-dependent rather than specific to the approximator. A third, larger vorticity--streamfunction problem shows the same ordering: the residual-optimal solver violates its structural identity by over six orders above tolerance, despite a residual margin of only 9.37%. Because a scalar loss cannot expose this, I introduce a lexicographic admissibility gate spanning the structural identity, boundary trace, and a solvability integral that must vanish independently of the equation residual. All three must pass before residuals can compete. This gate catches three artifact classes but misses a fourth: a prescribed-structure prior yields fields that pass every single-run check, yet deleting the source term reveals that 97\% of the reported structure survives removal of the physics. Reference data and figures accompany the paper.
Comments11 pages, 6 pages,
