发表机构
The University of Manchester; Wuhan University（曼彻斯特大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出将离散能量作为有限元代理模型的无标签训练目标，通过理论推导与数值验证证明其有效性，为线性弹性静力学代理模型训练提供了新方法。
AI 中文摘要
有限元（FE）代理模型的监督训练需要参考解，而每个参考解是通过求解代理模型所要替代的系统得到的。组装后的离散势能提供了一种无需参考解的训练信号。本笔记通过证明记录了使该信号对于线性弹性静力学精确的恒等式：预测的能量与参考解的能量之差等于二分之一的刚度范数误差的平方，且能量的梯度等于刚度加权误差。因此，无标签离散能量最小化与刚度范数下的监督回归具有相同的唯一极小值点，且在每一点处的梯度都相同。围绕这一核心结果，本笔记还阐述了一个条件引理（将位移误差用能量间隙界定）、一个模式收缩恒等式（解释为何欧几里得位移误差不适合作为主要度量）、一个控制共轭梯度对代理模型预测后处理的切比雪夫界，以及一个针对联合嵌入预测架构（JEPA）在共享刚度算子上预训练的条件潜在分离命题，同时给出了一个明确的数值反例来限定其适用范围。所有带数值内容的断言均被实现为可执行的证伪检查；这些检查在合成测试问题和预注册实验运行验证拆分的16个实例组成的探测集上各执行了两次，所有不等式均成立，且报告了测得的紧度。最后一节解释了为何该构造无法通过直接最小化作用泛函扩展到弹性动力学，以及哪种时间离散形式能恢复精确性。
英文摘要
Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.
Comments9 pages. Theory note of the FE-JEPA programme
