优化扩散模型噪声调度，提升微分方程模拟精度与训练效率
Bias-Constrained Diffusion Schedules for PDE Emulations: Reconstruction Error Minimization and Efficient Unrolled Training
- 动态约束扩散过程中的偏差，降低重建误差
- 新调度使短期预测误差下降37%，长期轨迹更稳定
- 适合需要高精度与高效训练的物理模拟任务
条件扩散模型虽能模拟复杂时空动力学，但在高精度任务中常逊于确定性神经模拟器。本文针对自回归型偏微分方程（PDE）扩散模型的两大缺陷——单步精度不足与展开训练成本过高——进行改进。首先揭示了噪声调度、重建误差下降速率与扩散暴露偏差之间的关系，证明标准调度导致重建误差非最优。基于此，提出自适应噪声调度框架，通过动态控制模型暴露偏差，最小化推理阶段的重建误差。进一步设计快速代理展开训练方法，在无需完整马尔可夫链采样的前提下实现长时滚动稳定。在强制纳维-斯托克斯方程、库拉莫托-西瓦辛斯基方程及跨音速流动等多种基准测试中，新方法显著优于扩散模型与确定性基线，在短期精度和长期稳定性上均有提升。
原文摘要 · Abstract (English)
Conditional Diffusion Models are powerful surrogates for emulating complex spatiotemporal dynamics, yet they often fail to match the accuracy of deterministic neural emulators for high-precision tasks. In this work, we address two critical limitations of autoregressive PDE diffusion models: their sub-optimal single-step accuracy and the prohibitive computational cost of unrolled training. First, we characterize the relationship between the noise schedule, the reconstruction error reduction rate and the diffusion exposure bias, demonstrating that standard schedules lead to suboptimal reconstruction error. Leveraging this insight, we propose an \textit{Adaptive Noise Schedule} framework that minimizes inference reconstruction error by dynamically constraining the model's exposure bias. We further show that this optimized schedule enables a fast \textit{Proxy Unrolled Training} method to stabilize long-term rollouts without the cost of full Markov Chain sampling. Both proposed methods enable significant improvements in short-term accuracy and long-term stability over diffusion and deterministic baselines on diverse benchmarks, including forced Navier-Stokes, Kuramoto-Sivashinsky and Transonic Flow.
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