用谱域注意力融合物理残差，让扩散PDE求解器更快更稳。
PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attention
- 在频域通过注意力机制直接嵌入物理残差，无需梯度优化。
- 仅需10到100步去噪，推理速度提升15至250倍。
- 适合高噪声场景下的快速高精度PDE求解，无需调参。
基于扩散模型的偏微分方程（PDE）求解器常受限于依赖梯度的测试时优化过程，且易受优化不稳定性影响，难以应对噪声残差。本文提出PRISMA（PDE残差感知的谱域调制注意力），将PDE残差以类注意力机制嵌入模型的频域架构中，实现无梯度优化的推理。相比以往仅将残差作为外部损失目标的方法，PRISMA将其作为内在结构特征，具备固有的高速、鲁棒、准确与免调参特性。在五个基准PDE任务上，尤其在含噪声观测下，其精度与先前方法相当，但推理成本显著降低：去噪步数减少10至100倍，推理速度提升15至250倍。
原文摘要 · Abstract (English)
Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance. They additionally suffer from optimization instabilities and are unable to dynamically adapt their inference scheme in the presence of noisy PDE residuals. To address these limitations, we introduce PRISMA (PDE Residual Informed Spectral Modulation with Attention), a conditional diffusion neural operator that embeds PDE residuals directly into the model's architecture via an attention-inspired modulation mechanism in the spectral domain, enabling gradient-descent free inference. In contrast to previous methods that use PDE loss solely as external optimization targets, PRISMA integrates PDE residuals as integral architectural features, making it inherently fast, robust, accurate, and free from sensitive hyperparameter tuning. We show that PRISMA has competitive accuracy, at substantially lower inference costs, compared to previous methods across five benchmark PDEs especially with noisy observations, while using 10x to 100x fewer denoising steps, leading to 15x to 250x faster inference.
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