Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

2026-05-11

AI Summary

This work addresses the instability in long-term forecasting of time-dependent partial differential equations (PDEs), which arises from error accumulation in autoregressive models and dynamic drift. To mitigate these issues, the authors propose the Structured Spectral Propagator (SSP) framework, which operates through an analysis–propagation–synthesis pipeline in a latent space. SSP decouples spatial details from recurrent dynamics and introduces an explicit, structured spectral propagation mechanism endowed with strong inductive biases. This mechanism integrates a frequency-conditioned linear backbone with a nonlinear spectral closure model to enable stable and high-fidelity long-term evolution. Experimental results demonstrate that SSP significantly outperforms existing methods on long-horizon extrapolation tasks, achieving up to a 48.9% reduction in relative L2 error, and exhibits exceptional predictive stability and generalization capability.

Abstract

Long-horizon forecasting of time-dependent partial differential equations (PDEs) is critical for characterizing the sustained evolution of physical systems. While neural operators have emerged as efficient surrogates, they typically learn implicit finite-time transitions from discrete observations. When deployed autoregressively, such propagators often suffer from rapid error accumulation and dynamic drift. To address this, we propose a neural forecasting framework that reformulates PDE rollout as learning a Structured Spectral Propagator (SSP) in a propagation-oriented latent space. Following an analysis-propagation-synthesis design, our framework: (i) maps physical states into a shared, time-consistent spatial representation; (ii) projects this space into a compact propagation state to isolate recurrent dynamics from fine-grained spatial details, thereby decoupling reconstruction fidelity from rollout regularity; and (iii) evolves retained spectral modes using a frequency-conditioned linear backbone complemented by a nonlinear spectral closure to account for truncated interactions. This explicit structuring endows the propagator with a strong inductive bias for coherent modal evolution. Extensive experiments demonstrate that SSP significantly outperforms state-of-the-art baselines, reducing relative $L_2$ errors by up to 48.9% and exhibiting improved stability in temporal extrapolation beyond the supervised horizon.

Problem

• long-horizon forecasting
• partial differential equations
• error accumulation
• dynamic drift
• neural operators

Innovation

• Structured Spectral Propagator
• neural operators
• long-horizon PDE forecasting
• spectral dynamics
• inductive bias