发表机构
Louisiana Tech University; Texas State University（路易斯安那理工大学; 得克萨斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过统一对比实验发现，已知方程残差约束在相同调参预算下优于通用正则化器，其优势特定于非线性PDE，且不依赖数据稀疏性，但在网格欠分辨率下强制执行会引入偏差。
AI 中文摘要
神经PDE代理模型日益融入结构先验，但尚不清楚其增益究竟源于物理特定信息，还是仅仅来自正则化和训练选择。我们在统一协议下，将若干此类先验与匹配的从零训练的神经算子基线进行比较。我们的核心结果是，在相同调参预算下，已知方程残差始终优于最佳通用正则化器。在固定容量下，这一优势在线性和非线性PDE中均出现，但容量扫描揭示了一个显著区别：对于Burgers、KdV和Allen-Cahn方程，优势持续存在并增长，而对于线性热传导和对流扩散方程，优势则降至持平或更低。因此，残差的持久价值特定于非线性算子。我们进一步推翻了一个预先注册的假设，即该优势仅由数据稀疏性激活：即使在完全监督下，残差仍然具有优势。然而，其有效性存在明确边界。在网格欠分辨率下，非线性粗场不再满足朴素控制方程残差，强制执行该残差反而有害。相比之下，跨族预训练和上下文条件化在研究的场景中未能超越强大的从零训练基线。这些结果共同揭示了已知物理规律何时为神经PDE模型提供非冗余信息、何时不提供，以及何时强制执行会引入偏差。
英文摘要
Neural PDE surrogates increasingly incorporate structural priors, yet it is often unclear whether their gains arise from physics-specific information or simply from regularization and training choices. We evaluate several such priors under a common protocol against a matched from-scratch neural operator baseline. Our central result is that a known-equation residual consistently outperforms the best generic regularizer at equal tuning budget. At fixed capacity this benefit appears across linear and nonlinear PDEs, but a capacity sweep reveals a sharp distinction: the advantage persists and grows for Burgers, KdV, and Allen-Cahn, while collapsing toward or below parity for linear heat and advection-diffusion. Thus, the durable value of the residual is specific to nonlinear operators. We further falsify a pre-registered hypothesis that the benefit is activated only by data sparsity: the residual remains advantageous even under full supervision. Its usefulness does, however, have a clear boundary. Under grid under-resolution, nonlinear coarse fields no longer satisfy the naive governing-equation residual, and enforcing it becomes actively harmful. In contrast, cross-family pretraining and in-context conditioning fail to outperform the strong from-scratch baseline in the regime studied. Together, these results identify when known physics provides non-redundant information to neural PDE models, when it does not, and when enforcing it introduces bias.
