Research map / Graph learning / Knowledge and temporal graphs
PURE: Prompt Evolution with Graph ODE for Out-of-distribution Fluid Dynamics Modeling
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Proposes PURE, which learns time-evolving prompt embeddings via a graph ODE with observation interpolation, plus mutual-information minimization, to adapt fluid dynamics forecasting models to out-of-distribution parameter and temporal shifts.
Problem
Out-of-distribution fluid dynamics forecasting under parameter-based shifts (varying PDE coefficients) and temporal distribution shifts during autoregressive rollout.
Approach
Initializes prompt embeddings from multi-view context (attention plus frequency-domain FFT processing of observations and system parameters), evolves them with a graph ODE whose interactions are attention-weighted by interpolated observations, and concatenates them with observation embeddings while minimizing their mutual information via a Jensen-Shannon estimator.
Key results
- PURE reaches MSE 0.0323 (in-distribution) and 0.0328 (OOD) on Prometheus, improving on the second-best DGODE by 6.10% and 8.63%.
- On Navier-Stokes, PURE gets MSE 0.0752 ID and 0.0763 OOD, a 26.07% OOD improvement over the second-best model.
- Ablation on S-SWE shows relative L2 error rising from 0.1357 (full PURE) to 0.1882 without graph ODE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply PURE | fluid dynamics forecasting · Prometheus w/ OOD | MSE: 0.0359 -> 0.0328 | Helped |
| Apply PURE | fluid dynamics forecasting · Navier–Stokes w/ OOD | MSE: 0.0925 -> 0.0763 | Helped |
| Apply PURE | fluid dynamics forecasting · Spherical-SWE w/ OOD | MSE: 0.0028 -> 0.0024 | Helped |
| Apply PURE | fluid dynamics forecasting · 3D Reaction-Diffusion w/ OOD | MSE: 0.0156 -> 0.0127 | Helped |
| Apply PURE | fluid dynamics forecasting · ERA5 w/ OOD | MSE: 0.0563 -> 0.0401 | Helped |
| Add Graph ODE instead of static prompt embeddings (ablation) | PURE · S-SWE | Relative L2 error: 0.1882 -> 0.1357 | Helped |
| Add interpolation to the Graph ODE (ablation) | PURE · S-SWE | Relative L2 error: 0.1696 -> 0.1357 | Helped |
| Add mutual information minimization (ablation) | PURE · S-SWE | Relative L2 error: 0.1588 -> 0.1357 | Helped |
| Add frequency-domain enhancement (FFT) (ablation) | PURE · S-SWE | Relative L2 error: 0.1602 -> 0.1357 | Helped |
| Add Graph ODE (ablation) | FNO + PURE · Navier–Stokes | MSE: 0.1282 -> 0.0987 | Helped |
| Add interpolation (ablation) | FNO + PURE · Navier–Stokes | MSE: 0.1097 -> 0.0987 | Helped |
| Add mutual information minimization (ablation) | FNO + PURE · Navier–Stokes | MSE: 0.1182 -> 0.0987 | Helped |
Limitations
- Assumes training and test data are IID, which may fail under extreme environmental changes.
- May struggle with high-dimensional, complex fluid systems.
- Acknowledged high computational complexity and resource needs.
Experimental setup
Datasets: Prometheus 2D Navier-Stokes Spherical shallow water equations 3D reaction-diffusion ERA5
Benchmarks: Prometheus Navier-Stokes Spherical-SWE 3D Reaction-Diffusion ERA5
Metrics: MSE relative L2 error
Baselines: U-Net ResNet ViT SwinT FNO UNO CNO NMO CGODE DGODE LEADS CODA
Builds on: DGODE Fourier neural operator prompt learning graph ODE
Closest papers by meaning
- Prometheus: Out-of-distribution Fluid Dynamics Modeling with Disentangled Graph ODE (ICML 2024 · similarity 0.88)
- FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation (NeurIPS 2025 · similarity 0.86)
- Simulation-Free Training of Neural ODEs on Paired Data (NeurIPS 2024 · similarity 0.86)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.85)
- Mesh-Free Convolution: Learned Spectral Attenuation and Transport (NeurIPS 2026 · similarity 0.85)
- Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks (ICLR 2025 · similarity 0.85)
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