# Derived Fields Preserve Fine-Scale Detail in Budgeted Neural Simulators

arXiv: https://arxiv.org/abs/2603.29224 · HTML: https://mlsanity.com/paper/2603.29224/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 50 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 26 of 26)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Variational image compression with a scale hyperprior](https://mlsanity.com/paper/1802.01436/) (2018)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [A Neural PDE Solver with Temporal Stencil Modeling](https://mlsanity.com/paper/2302.08105/) (2023)
- [Pi-fusion: Physics-informed diffusion model for learning fluid dynamics](https://mlsanity.com/paper/2406.03711/) (2024)
- [HelmFluid: Learning Helmholtz Dynamics for Interpretable Fluid Prediction](https://mlsanity.com/paper/2310.10565/) (2023)
- [LOGLO-FNO: Efficient Learning of Local and Global Features in Fourier Neural Operators](https://mlsanity.com/paper/2504.04260/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Learning Generalized Quasi-Geostrophic Models Using Deep Neural Numerical Models](https://mlsanity.com/paper/1911.08856/) (2019)

## Abstract (arXiv)

Fine-scale-faithful neural simulation under fixed storage budgets remains challenging. Many existing methods reduce high-frequency error by improving architectures, training objectives, or rollout strategies. However, under budgeted coarsen-quantize-decode pipelines, fine detail can already be lost when the carried state is constructed. In the canonical periodic incompressible Navier-Stokes setting, we show that primitive and derived fields undergo systematically different retained-band distortions under the same operator. Motivated by this observation, we formulate Derived-Field Optimization (DerivOpt), a general state-design framework that chooses which physical fields are carried and how storage budget is allocated across them under a calibrated channel model. Across the full time-dependent forward subset of PDEBench, DerivOpt not only improves pooled mean rollout nRMSE, but also delivers a decisive advantage in fine-scale fidelity over a broad set of strong baselines. More importantly, the gains are already visible at input time, before rollout learning begins. This indicates that the carried state is often the dominant bottleneck under tight storage budgets. These results suggest a broader conclusion: in budgeted neural simulation, carried-state design should be treated as a first-class design axis alongside architecture, loss, and rollout strategy.