# Towards Multi-spatiotemporal-scale Generalized PDE Modeling

arXiv: https://arxiv.org/abs/2209.15616 · HTML: https://mlsanity.com/paper/2209.15616/

## Key facts

- Cited by 197 papers in the map (data through 14 Oct 2026).
- Builds on 81 of its 144 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Builds on (most cited first, 30 of 81)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [A Survey on Deep Learning in Medical Image Analysis](https://mlsanity.com/paper/1702.05747/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)

## Built on by (most cited first, 30 of 197)

- [ClimaX: A foundation model for weather and climate](https://mlsanity.com/paper/2301.10343/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [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)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [ClimateLearn: Benchmarking Machine Learning for Weather and Climate Modeling](https://mlsanity.com/paper/2307.01909/) (2023)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [Walrus: A Cross-Domain Foundation Model for Continuum Dynamics](https://mlsanity.com/paper/2511.15684/) (2025)
- [Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers](https://mlsanity.com/paper/2405.17527/) (2024)
- [PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics](https://mlsanity.com/paper/2409.09811/) (2024)
- [PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations](https://mlsanity.com/paper/2505.24717/) (2025)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)

## Abstract (arXiv)

Partial differential equations (PDEs) are central to describing complex physical system simulations. Their expensive solution techniques have led to an increased interest in deep neural network based surrogates. However, the practical utility of training such surrogates is contingent on their ability to model complex multi-scale spatio-temporal phenomena. Various neural network architectures have been proposed to target such phenomena, most notably Fourier Neural Operators (FNOs), which give a natural handle over local & global spatial information via parameterization of different Fourier modes, and U-Nets which treat local and global information via downsampling and upsampling paths. However, generalizing across different equation parameters or time-scales still remains a challenge. In this work, we make a comprehensive comparison between various FNO, ResNet, and U-Net like approaches to fluid mechanics problems in both vorticity-stream and velocity function form. For U-Nets, we transfer recent architectural improvements from computer vision, most notably from object segmentation and generative modeling. We further analyze the design considerations for using FNO layers to improve performance of U-Net architectures without major degradation of computational cost. Finally, we show promising results on generalization to different PDE parameters and time-scales with a single surrogate model. Source code for our PyTorch benchmark framework is available at https://github.com/microsoft/pdearena.