Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

TL;DR

This paper introduces a novel neural forecasting framework for PDEs that uses a structured spectral propagator in a latent space, significantly improving long-horizon prediction accuracy and stability.

Contribution

It proposes a new spectral propagator approach in a latent space for PDE forecasting, addressing error accumulation and dynamic drift issues.

Findings

SSP reduces relative L2 errors by up to 48.9%.

It outperforms state-of-the-art baselines in long-horizon PDE forecasting.

The method shows improved stability in temporal extrapolation.

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.