将微分方程约束融入去噪器，提升生成解的物理一致性。
Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations
- 在去噪器中加入微分方程的软性先验，而非强加约束
- 相比标准方法，生成解更符合物理规律且对模型误差更鲁棒
- 适合需要高物理保真度的科学计算场景
扩散模型已成为偏微分方程（PDE）解的强大生成先验。现有方法通过将PDE残差作为损失正则项或在推理时调整来施加物理约束，但会偏离真实数据分布，尤其当控制PDE存在误设时问题更严重。为克服此问题并充分利用PDE约束，我们从PDE推导出软性归纳偏置，并将其嵌入去噪器架构。实验表明，这种软约束去噪器能有效利用约束知识提升解的合规性，同时保留足够灵活性，在数据与模型不一致时仍可适当偏离约束。
原文摘要 · Abstract (English)
Diffusion models have become a powerful generative prior for solutions of partial differential equations (PDEs). Existing approaches enforce physical constraints either by adding the PDE residuals as loss regularizers or through inference-time adjustments. These methods bias the model away from the true data distribution, which is especially problematic when the governing PDE is misspecified. To circumvent these issues while making the most out of the PDE constraint, we introduce soft inductive biases into the denoiser architecture derived from the PDEs. We show that these softly constrained denoisers exploit constraint knowledge to improve compliance over standard denoisers, while maintaining enough flexibility to deviate from it in case of misspecification with respect to observed data.
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