# Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs

arXiv: https://arxiv.org/abs/2608.18148 · HTML: https://mlsanity.com/paper/2608.18148/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [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)
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- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Learning Dissipative Dynamics in Chaotic Systems](https://mlsanity.com/paper/2106.06898/) (2021)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.01174/) (2023)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure](https://mlsanity.com/paper/2105.05690/) (2021)
- [Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures](https://mlsanity.com/paper/2105.14410/) (2021)
- [Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds](https://mlsanity.com/paper/2109.00700/) (2021)
- [Comparison of neural closure models for discretised PDEs](https://mlsanity.com/paper/2210.14675/) (2022)

## Abstract (arXiv)

We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, we propose a spatiotemporal neural operator mapping a resolved history slab on Omega×[-Tin,0] to a resolved future slab on Omega×[0,Tout]. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length Tin via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.