Problem
Research questions and friction points this paper is trying to address.
Neural Operators
PDE Forecasting
Error Accumulation
Fine-scale Structures
F$^3$NO: Frequency-Decomposed Finite-Time Flow-map Neural Operators with Cross-Scale Conditioning
🤖 AI Summary
This study addresses the substantial error accumulation and limited capacity to resolve fine-grained structures encountered by neural operators when predicting partial differential equations (PDEs). To this end, we propose a frequency-decomposed finite-time flow map neural operator. The method introduces a cross-scale conditioning mechanism in which low-frequency features guide high-frequency nonlinear refinement, enabling direct prediction of states at specified time instants while dynamically adjusting branch contributions. Furthermore, piecewise-parallel and recursive propagation strategies are integrated to optimize long-trajectory accuracy. Evaluated across five PDE benchmarks, the proposed model significantly outperforms both autoregressive and direct-prediction baselines with fewer parameters, demonstrating substantially improved predictive accuracy.
📝 Abstract
Neural operators enable fast PDE forecasting, but repeated predictions accumulate errors and fine-scale structures remain difficult to resolve. We introduce a frequency-decomposed finite-time flow-map neural operator (F$^3$NO) that leverages updated low-frequency features to guide nonlinear refinement of high-frequency information. Within each layer, this cross-scale conditioning connects global spectral processing with local detail refinement. The model directly predicts states at specified future times and adjusts the contributions of the two branches according to the prediction interval. For longer trajectories, it combines parallel predictions within short temporal segments with recursive propagation between segments. Experiments on five PDE benchmarks demonstrate improved forecasting accuracy over autoregressive and direct-prediction baselines. Ablations show that frequency-decomposed refinement can improve accuracy with fewer parameters, while the benefits of segmentation depend on spatial resolution and dynamical regime.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Neural Operators
Frequency Decomposition
Cross-Scale Conditioning
Finite-Time Flow-Map
PDE Forecasting
🔎 Similar Papers
No similar papers found.
