# Coarse Graining with Neural Operators for Simulating Chaotic Systems

arXiv: https://arxiv.org/abs/2408.05177 · HTML: https://mlsanity.com/paper/2408.05177/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2024.

## Built on by (most cited first, 6 of 6)

- [TensorGRaD: Tensor Gradient Robust Decomposition for Memory-Efficient Neural Operator Training](https://mlsanity.com/paper/2501.02379/) (2025)
- [Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators](https://mlsanity.com/paper/2506.19396/) (2025)
- [An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence](https://mlsanity.com/paper/2509.20365/) (2025)
- [From Black Hole to Galaxy: Neural Operator: Framework for Accretion and Feedback Dynamics](https://mlsanity.com/paper/2512.01576/) (2025)
- [Resolution-Agnostic Neural Operators for Multi-Rate Sparse-View CT](https://mlsanity.com/paper/2512.12236/) (2025)
- [Pushing the Limits of High-Resolution Weather Forecasting through Data Scaling](https://mlsanity.com/paper/2608.14652/) (2026)

## Abstract (arXiv)

Accurately predicting the long-term behavior of chaotic systems is crucial for various applications such as climate modeling. However, achieving such predictions typically requires iterative computations over a dense spatiotemporal grid to account for the unstable nature of chaotic systems, which is expensive and impractical in many real-world situations. An alternative approach to such a full-resolved simulation is using a coarse grid and then correcting its errors through a textitclosure model, which approximates the overall information from fine scales not captured in the coarse-grid simulation. Recently, ML approaches have been used for closure modeling, but they typically require a large number of training samples from expensive fully-resolved simulations (FRS). In this work, we prove an even more fundamental limitation, i.e., the standard approach to learning closure models suffers from a large approximation error for generic problems, no matter how large the model is, and it stems from the non-uniqueness of the mapping. We propose an alternative end-to-end learning approach using a physics-informed neural operator (PINO) that overcomes this limitation by not using a closure model or a coarse-grid solver. We first train the PINO model on data from a coarse-grid solver and then fine-tune it with (a small amount of) FRS and physics-based losses on a fine grid. The discretization-free nature of neural operators means that they do not suffer from the restriction of a coarse grid that closure models face, and they can provably approximate the long-term statistics of chaotic systems. In our experiments, our PINO model achieves a 330x speedup compared to FRS with a relative error ~ 10%. In contrast, the closure model coupled with a coarse-grid solver is 60x slower than PINO while having a much higher error ~186% when the closure model is trained on the same FRS dataset.