# TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations

arXiv: https://arxiv.org/abs/2606.21189 · HTML: https://mlsanity.com/paper/2606.21189/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 19 Jun 2026.

## Builds on (most cited first, 18 of 18)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Lightweight LLM Agent Memory with Small Language Models](https://mlsanity.com/paper/2604.07798/) (2026)
- [Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena](https://mlsanity.com/paper/2504.20249/) (2025)
- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)
- [PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils](https://mlsanity.com/paper/2509.26186/) (2025)
- [Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings](https://mlsanity.com/paper/2602.16193/) (2026)
- [Geometric Neural Operators via Lie Group-Constrained Latent Dynamics](https://mlsanity.com/paper/2602.16209/) (2026)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)
- [From Similarity to Structure: Training-free LLM Context Compression with Hybrid Graph Priors](https://mlsanity.com/paper/2604.23277/) (2026)

## Abstract (arXiv)

Non-stationary partial differential equations (PDEs) arise throughout scientific computing, where the dominant frequency content and energy distribution can drift over time. While efficient in PDE solving, many spectral neural operators apply a shared spectral response across rollout stages, leading to mismatch with time-varying spectra in non-stationary systems. To address this issue, we propose Time-Frequency Gated Spectral Neural Operator (TF-SNO), a state-adaptive framework with learnable time-frequency gating inside spectral blocks. TF-SNO extracts compact frequency-domain and physical-space statistics from the current state to generate modulation coefficients, enabling the spectral response to evolve with the dynamics. TF-SNO learns temporal variation implicitly from the evolving state without introducing an explicit time dimension or time embedding, keeping the modeling complexity low. We further embed the adaptive operator blocks to accurately capture the multi-scale features, thereby improving long-horizon stability. Experiments on six non-stationary PDE benchmarks in 1D and 2D demonstrate that TF-SNO significantly reduces prediction errors and improves robustness compared to strong baselines, with particularly clear gains in long rollout, suggesting the effectiveness of state-dependent spectral adaptation in modeling non-stationary physical systems.