精确性的极限：论自动微分在物理信息机器学习中的失效
The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning
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中文总结 AI 辅助
本文指出自动微分虽能提供机器精度导数，但缺乏物理结构意识，导致数值精确却物理错误的导数，涵盖对流、扩散、守恒等广泛问题，并据此重新定位下一代PDE神经代理模型的构建方向。
中文摘要 AI 辅助
自动微分（AD）使神经网络能够以机器精度计算控制方程的导数，而这一精度使其成为物理信息机器学习的计算支柱。然而，数学意义上的精确性并不等同于对物理的忠实性。本文论证，一个导数在数值上可以是完美的，但仍然可能是针对当前问题错误的导数，因为自动微分在构造上不具备解必须遵循的物理结构的概念。对流及其相关的方向性、扩散和色散，只是涵盖计算科学与工程所有分支的更长列表中最显眼的实例，该列表还包括守恒、热力学一致性、对称性、辛结构、正性、单调性和有界性。认识到这一更广泛的差距，重新定义了该领域应如何构建下一代由偏微分方程驱动的神经代理模型。
英文摘要
Automatic differentiation (AD) lets neural networks compute derivatives of governing equations to machine precision, and this precision has made it the computational backbone of physics-informed machine learning. Yet exactness in the mathematical sense is not the same as fidelity to the physics. Here I argue that a derivative can be numerically perfect and still be the wrong derivative for the problem at hand, because AD, by construction, has no notion of the physical structure a solution must obey. Convection and its associated directionality, diffusion, and dispersion are only the most visible instances of a much longer list that spans all branches of computational science and engineering, including conservation, thermodynamic consistency, symmetry, symplectic structure, positivity, monotonicity, and boundedness. Recognizing this broader gap reframes how the field should build the next generation of PDE-driven neural surrogates.
