Bias-Constrained Diffusion Schedules for PDE Emulations: Reconstruction Error Minimization and Efficient Unrolled Training

TL;DR

This paper introduces an adaptive noise schedule and proxy unrolled training to improve the accuracy and efficiency of PDE emulation using diffusion models, outperforming existing methods on various benchmarks.

Contribution

The authors propose a novel adaptive noise schedule and proxy unrolled training framework that enhance PDE emulation accuracy and reduce computational costs.

Findings

Significant accuracy improvements over baseline diffusion models.

Enhanced long-term stability in PDE simulations.

Effective on diverse benchmarks like Navier-Stokes and Kuramoto-Sivashinsky.

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.