PDE训练输入的损耗压缩：场重建误差不决定训练后算子的成本

中文摘要 AI 辅助

算子学习基准以全精度存储，且已增长至太字节规模。率失真理论规定了存储场需要多少比特，而实践者需要知道在压缩数据上训练的算子将有多准确。我们表明前者并不决定后者，并测量了其原因，即压缩输入场而目标和测试输入保持全精度。解算子会衰减其输入的扰动。将压缩场通过一个已在全精度下训练的代理模型，可测量该代理模型传递了多少扰动。该比例与底层方程的平滑行为一致，且在PDE族之间跨越两个数量级以上。场重建误差在衰减之前计算，无法看到衰减。对于以均方误差训练的算子，它在数据集上的104次成本比较中反转了36次，而由相同前向传递构建的探针则反转了12次。PDEBench以相同初始条件存储的两个族，在相同场误差下，下游差异为三倍。在参考方案的相对L2目标下，分离度缩小，而族级中位传递因子的排序不变。在一次全精度训练运行后，探针仅通过前向传递评估整个率曲线。它在我们测试的编解码器和架构上一致地对数据集和率进行排序，而其幅度在它们之间不转移。

英文摘要

Operator-learning benchmarks ship as full-precision arrays, and they have grown to terabyte scale. A curator who wants to distribute one has to decide how coarsely to store it. That decision is usually made by fixing a tolerance on the reconstruction error of the stored field. We show that this quantity is measured in the wrong place. It compares the stored field with the original, before any model is trained. What the curator is buying is the accuracy of a model trained on the compressed copy, and the two can disagree: two PDEBench families stored to identical field error differ threefold downstream. A solution operator smooths, so only part of the codec error ever reaches the model's output. That part can be measured. Push a compressed field through a model already trained at full precision, and compare its output against the output on the clean field. The measurement costs forward passes and no training on compressed data. It spans more than two orders of magnitude across six PDE families, and it orders downstream cost where field error does not, inverting 12 of 104 cross-dataset comparisons against field error's 36. Read as a budget, the same curve nominates a storage rate for each dataset. A back-check against trained models finds those rates wrong in both directions by up to a factor of two in stored bits.

发表机构

• Powermore Ltd.（Powermore 有限公司）

机构由 AI 辅助整理，请以论文原文为准。