# Do Neural PDE Solvers Learn the Right Dynamics?

arXiv: https://arxiv.org/abs/2610.06952 · HTML: https://mlsanity.com/paper/2610.06952/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 3 Oct 2026.

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

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Generative AI for fast and accurate statistical computation of fluids](https://mlsanity.com/paper/2409.18359/) (2024)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Turbulence in Focus: Benchmarking Scaling Behavior of 3D Volumetric Super-Resolution with BLASTNet 2.0 Data](https://mlsanity.com/paper/2309.13457/) (2023)
- [Dynamical errors in machine learning forecasts](https://mlsanity.com/paper/2504.11074/) (2025)
- [TRIE: An Evaluation Framework for Stochastic PDE Surrogates](https://mlsanity.com/paper/2607.00196/) (2026)

## Abstract (arXiv)

Neural PDE solvers can achieve low prediction errors, but do they reproduce the dynamics of the systems they model? Prediction scores alone offer an incomplete answer: they measure agreement with reference solutions but provide limited insight into how errors accumulate, nearby states diverge, or extreme events arise. We propose an evaluation framework that directly examines these behaviors in deterministic and stochastic neural solvers. By evolving ensembles of nearby initial states and comparing them with direct numerical simulation, we assess three complementary aspects of learned dynamics: error formation, ensemble geometry, and extreme events. Experiments on two-dimensional Kolmogorov flow reveal limitations that conventional scores can obscure. Smaller trajectory errors can reflect weaker error amplification despite less accurate local updates. Models can match an ensemble's overall spread and effective dimension while failing to capture the spatial directions where nearby states diverge. Similarly, matching overall event frequencies can conceal failures to predict persistent extreme events. These findings show that improved prediction accuracy does not necessarily imply greater dynamical fidelity. Our framework makes this distinction measurable, providing concrete criteria for evaluating whether advances in neural PDE solvers better capture the underlying dynamics.