物理信息神经网络中的导数保真失效模式：来自函数值训练的强化基准证据
A derivative-fidelity failure mode in physics-informed neural networks: strengthened benchmark evidence from function-value training
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中文总结 AI 辅助
本文提出物理信息神经网络存在导数保真失效模式，通过仅训练函数值的基准实验证明，视觉上准确的函数逼近可能伴随显著更大的二阶导数误差，并提供诊断协议以区分数值精度与物理残差可靠性。
中文摘要 AI 辅助
物理信息神经网络（PINNs）利用自动微分来施加微分方程残差，但函数值上的良好一致性并不必然意味着导数的准确性。本文将导数保真度定义为PINNs的一种失效模式，并使用一维基准进行测试。多层感知器仅针对sin(x)和exp(x)的函数值进行训练，而通过自动微分获得的二阶导数则被单独评估。通过额外针对训练点密度、激活函数、端点密集评估以及L2和最大误差诊断的测试，该假设得到强化。结果表明，视觉上准确的函数逼近可以与显著更大的二阶导数误差共存，尤其是在高曲率边界区域附近。该实验提供了一种诊断协议，用于区分数值精度与物理残差可靠性。
英文摘要
Physics-informed neural networks (PINNs) use automatic differentiation to impose differential-equation residuals, but good agreement in function values does not necessarily imply accurate derivatives. This paper formulates derivative fidelity as a failure mode of PINNs and tests it with one-dimensional benchmarks. Multilayer perceptrons are trained only on function values for sin(x) and exp(x), while second derivatives obtained by automatic differentiation are evaluated separately. The hypothesis is strengthened by additional tests over training-point density, activation functions, endpoint-dense evaluation, and both L2 and maximum-error diagnostics. The results show that visually accurate function approximation can coexist with substantially larger second-derivative errors, especially near high-curvature boundary regions. The experiment provides a diagnostic protocol for distinguishing value accuracy from physics-residual reliability.
发表机构
- Osaka-seikei University（大阪成蹊大学）
机构由 AI 辅助整理，请以论文原文为准。
