Bias-Constrained Diffusion Schedules for PDE Emulations: Reconstruction Error Minimization and Efficient Unrolled Training
2026-04-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study Conditional Diffusion Models used to simulate complex physical systems but notice these models often aren’t as precise for tasks needing high accuracy. They identify problems related to how noise is added during training and propose an Adaptive Noise Schedule that improves accuracy by adjusting this noise dynamically. They also develop a Proxy Unrolled Training method that helps the model stay stable over longer predictions without much extra computation. Their approaches lead to better short-term accuracy and long-term simulation stability compared to existing methods on various physics benchmarks.
Conditional Diffusion ModelsSpatiotemporal dynamicsPDE (Partial Differential Equations)Noise scheduleExposure biasUnrolled trainingMarkov Chain samplingNavier-Stokes equationsKuramoto-Sivashinsky equationTransonic Flow
Authors
Constantin Le Cleï, Nils Thürey, Xiaoxiang Zhu
Abstract
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.