Physics-informed neural networks (PINNs) are often selected by a single scalar loss even when the quantity of interest is more specific. We study a hybrid design in which the governing PDE residual remains automatic-differentiation (AD) based, while finite differences (FD) appear only in a weak auxiliary term that penalizes gradients of the sampled residual field. The FD term regularizes the residual field without replacing the PDE residual itself. We examine this idea in two stages. Stage 1 is a controlled Poisson benchmark comparing a baseline PINN, the FD residual-gradient regularizer, and a matched AD residual-gradient baseline. Stage 2 transfers the same logic to a three-dimensional annular heat-conduction benchmark (PINN3D), where baseline errors concentrate near a wavy outer wall and the auxiliary grid is implemented as a body-fitted shell adjacent to the wall. In Stage 1, the FD regularizer reproduces the main effect of residual-gradient control while exposing a trade-off between field accuracy and residual cleanliness. In Stage 2, the shell regularizer improves the application-facing quantities, namely outer-wall flux and boundary-condition behavior. Across seeds 0-5 and 100k epochs, the most reliable tested configuration is a fixed shell weight of 5e-4 under the Kourkoutas-beta optimizer regime: relative to a matched run without the shell term, it reduces the mean outer-wall BC RMSE from 1.22e-2 to 9.29e-4 and the mean wall-flux RMSE from 9.21e-3 to 9.63e-4. Adam with beta2=0.999 becomes usable when the initial learning rate is reduced to 1e-3, although its shell benefit is less robust than under Kourkoutas-beta. Overall, the results support a targeted view of hybrid PINNs: an auxiliary-only FD regularizer is most valuable when it is aligned with the physical quantity of interest, here the outer-wall flux.

翻译：物理信息神经网络（PINNs）常通过单一标量损失函数进行优化，即便实际关注的目标量具有更精细的物理特性。本文研究了一种混合设计：控制方程残差仍基于自动微分（AD），而有限差分（FD）仅以弱辅助项形式出现，用于惩罚采样残差场梯度。该FD项在不替代PDE残差本身的前提下，对残差场进行正则化。我们分两阶段验证该思想：第一阶段，在可控泊松问题基准测试中比较标准PINN、FD残差梯度正则化方法及匹配的AD残差梯度基线；第二阶段，将相同逻辑迁移至三维环形热传导基准（PINN3D），其中基线误差集中于波浪形外壁面附近，辅助网格采用贴附壁面的体拟合壳层实现。结果表明，在第一阶段，FD正则化复现了残差梯度控制的主要效果，但暴露出场精度与残差纯净度之间的权衡关系；在第二阶段，壳层正则化改善了外壁面热流与边界条件行为等面向应用的物理量。使用不同随机种子（0-5）及10万轮训练后，最可靠的配置为Kourkoutas-beta优化器配合固定壳层权重5e-4：相较于无壳层项的匹配实验，外壁面边界条件均方根误差（BC RMSE）从1.22e-2降至9.29e-4，壁面热流均方根误差从9.21e-3降至9.63e-4。当初始学习率降至1e-3时，采用beta2=0.999的Adam优化器亦可胜任，但其壳层收益的稳健性弱于Kourkoutas-beta策略。综合而言，实验结果支持混合PINNs的目标导向观点：当仅含辅助项的FD正则化与关注的物理量（此处为外壁面热流）对齐时，其价值最为显著。