发表机构
AGH University of Krakow（克拉科夫AGH科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究对比PINNs中FD与AD的导数计算性能，发现FD精度相当且速度更快、内存占用更少，其随机变体在稳态问题上优于AD，还解决了含样本间依赖架构的逐样本导数计算问题。
AI 中文摘要
我们系统研究了物理信息神经网络（PINNs）中作为自动微分（AD）替代方案的有限差分（FD）导数计算方法。在三个基准偏微分方程（PDE）上，我们表明，通过适当校准步长，FD在每个问题上的精度与AD相当，同时在所有测试的批量大小范围内运行更快，且占用的GPU内存显著更少；我们提出的随机变体在一个稳态问题上的性能优于AD。我们进一步表明，对于具有样本间依赖关系的神经架构（如批量归一化（BatchNorm）、自注意力机制），标准PyTorch自动微分用法存在隐性错误；在PINN相关批量大小下，正确的逐样本方法在计算上不可行，而FD提供了一种仅前向的近似，经验上与真实逐样本导数的接近程度达一个数量级。
英文摘要
We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.
Comments22 pages, 5 figures
