Autoregression-Free Neural Operators for Time-Dependent PDEs

2026-05-25

AI Summary

This work addresses the limitations of existing neural operators in solving time-dependent partial differential equations (PDEs), which typically rely on autoregressive roll-out predictions that suffer from error accumulation and degraded stability over long horizons. To overcome this, the authors propose a novel non-autoregressive approach that maps the temporal evolution of PDEs into a latent space, models the continuous-time vector field therein, and learns it via flow matching to enable end-to-end continuous long-time prediction. This method introduces, for the first time in neural operators, a non-autoregressive mechanism for continuous-time modeling while explicitly conditioning on physical parameters, substantially enhancing stability and generalization in long-horizon forecasts. Experiments across six PDE benchmarks demonstrate that the proposed approach significantly reduces roll-out errors and improves both simulation accuracy and robustness compared to current baselines.

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.

Problem

• time-dependent PDEs
• autoregressive rollout
• error accumulation
• long-horizon prediction
• neural operators

Innovation

• neural operators
• autoregression-free
• continuous-time modeling
• flow matching
• time-dependent PDEs