对称强制变分问题神经近似的无参考记录能量-神谕恢复：符合里泽重构与存档级选择
Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection
- Laboratory for Analysis and Modeling of Systems and Decision Support (LAMSAD), Hassan 1st University of Settat（塞塔特哈桑一世大学系统与决策支持分析建模实验室）
- Al Akhawayn University in Ifrane（伊夫兰阿赫拉万大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
针对对称强制变分问题的神经近似，提出无参考的记录能量神谕恢复准则，通过符合里泽监测器实现存档级选择，在扩散、弹性等问题上验证了其有效性。
中文摘要 AI 辅助
神经PDE训练会产生有限的检查点存档，但其记录的能量误差若无精确解则无法获取，而基于损失的选择不一定能恢复记录的能量神谕。针对对称强制变分问题的可容许神经近似，我们引入一种基于最小化可计算符合里泽监测器的无参考选择规则。精确残差能量恒等式与符合投影使该监测器成为无条件下界，在嵌套符合细化下单调收敛至每个记录的能量误差；在饱和条件下，分层富集产生可计算的上估计，从而形成上下界。关键发现是存档选择具有顺序敏感性：有限分辨率下未解决的依赖检查点的分量可逆转神谕-非神谕排序，仅靠逐检查点恢复不足。对于有限存档，我们证明了均匀恢复，即收敛至记录的神谕误差，且在无饱和时，辅助分辨率足够精细时可实现记录的神谕选择。在饱和条件下，该界给出可计算的近神谕界，且区间分离时可证明唯一的记录神谕选择。我们还对记录分辨率损失进行了界定，并证明在规定的比较轨迹上神谕包含性成立。所得准则用可计算的、与训练无关的训练后选择替代了不可访问的精确误差最小化，该选择基于固有能量误差尺度，仅需计算得到的候选解和变分问题。在扩散和弹性问题（包括非制造穿孔板）上的实验，证明了能量尺度校准、神谕级选择及适度的后处理成本。
英文摘要
Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual-energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower-upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle-non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, training-independent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost.
