Autoregression-Free Neural Operators for Time-Dependent PDEs
2026-05-25 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors developed a new method called Autoregression-Free Neural Operators (AFNO) to better predict the behavior of systems described by time-dependent partial differential equations (PDEs). Unlike previous approaches that predict step-by-step and can accumulate errors over time, AFNO learns the system's behavior in a simpler hidden space and models its continuous time evolution. This approach helps make longer-term predictions more accurate and stable. They tested AFNO on six different PDE problems and found it generally outperforms existing methods.
Neural operatorsPartial differential equationsTime-dependent PDEsAutoregressive rolloutLatent spaceContinuous-time vector fieldsFlow matchingLong-horizon predictionError accumulation
Authors
Jiaquan Zhang, Caiyan Qin, Haoyu Bian, Libin Cai, Yi Lu, Chaoning Zhang, Wei Dong, Yuanfang Guo, Yang Yang, Hen Tao Shen
Abstract
Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.