# Spatial Context or Memory? Sizing the Inputs of Neural Surrogates for Partially Observed Fields

arXiv: https://arxiv.org/abs/2610.12723 · HTML: https://mlsanity.com/paper/2610.12723/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 37 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 15 of 15)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [Sample Complexity of Kalman Filtering for Unknown Systems](https://mlsanity.com/paper/1912.12309/) (2019)
- [Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme](https://mlsanity.com/paper/2511.19390/) (2025)
- [Merging Memory and Space: A State Space Neural Operator](https://mlsanity.com/paper/2507.23428/) (2025)
- [FieldFormer: Locality-Aware Transformers for Spatio-Temporal Modeling on Sparse Sensor Networks](https://mlsanity.com/paper/2510.03589/) (2025)
- [Learning Neural Operators from Partial Observations via Latent Autoregressive Modeling](https://mlsanity.com/paper/2601.15547/) (2026)
- [System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting](https://mlsanity.com/paper/2603.25025/) (2026)
- [How Much Memory Do We Need? Adaptive Memory Gate for Neural Operators](https://mlsanity.com/paper/2606.13443/) (2026)

## Abstract (arXiv)

Neural surrogates that forecast a physical field from sparse, noisy sensors read a window of nearby sensors and a history of past snapshots. Practitioners choose both by comparing fitted models with each other. Such comparisons cannot tell a model's shortfall from missing information. We measure the trade-off between window and history against the Bayes-optimal error of the same readings. An exact Gaussian analysis shows that noisy or sparse observation of a field with local dynamics makes the optimal forecast read distant sensors only through spatial correlation. On a forced linear field with a known optimum, trained models that read only the window come within 1% of it. We then train such models on viscous Burgers flow and the chaotic Kuramoto-Sivashinsky (KS) equation, with every deciding comparison fixed before any test evaluation. In every field except KS, eight snapshots remove a smaller fraction of the one-snapshot error as the lead time, the number of steps to the forecast target, grows. In KS, for windows up to one correlation length, they remove a larger fraction. This lead-time inversion persists across sensor densities, three architectures and a two-dimensional domain. It shrinks as damping lowers the Lyapunov exponent. Damping also removes the linear instability of KS. We therefore tie the inversion to instability and leave chaos and linear amplification unseparated. For surrogate design the results give two rules. Size the window in correlation lengths of the field. Split a fixed input budget between window and history for the lead time the model will serve, because the best split moves toward wider windows as the lead time grows. In Burgers flow the split chosen at lead time one ends with 13.5 times the best error at lead time eight.