Research map / Probabilistic and scientific modeling / Differential equations and PDEs
CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs
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Proposes CALM-PDE, an encoder-decoder built from continuous convolutions with epsilon-neighborhood-constrained kernels and learnable query points, performing latent time-stepping with a Transformer for arbitrarily discretized time-dependent PDEs.
Problem
Solving time-dependent PDEs on arbitrarily (including irregularly) discretized spatial domains with neural surrogates that are both memory-efficient and discretization-agnostic; attention-based latent models are memory-heavy while CNNs need regular grids.
Approach
Extends deep parametric continuous convolution with learnable query points constrained to a percentile-based epsilon neighborhood, distance-softmax weighting, random Fourier features and per-query kernel modulation (FiLM-style), used as encoder/decoder around a latent Transformer processor trained end-to-end.
Key results
- On 1D Burgers' (nu=1e-3), CALM-PDE achieves relative L2 error 0.0246, lower than the best baseline LNO at 0.0309; on 2D Navier-Stokes (nu=1e-4) it reaches 0.0301 vs OFormer 0.0380.
- On 2D Navier-Stokes nu=1e-5 CALM-PDE (0.1033) is outperformed by LNO (0.0789) and F-FNO (0.0844), and on 3D Navier-Stokes it reaches 0.6761 vs F-FNO 0.6466.
- CALM-PDE is significantly faster than OFormer and LNO in inference and uses less training memory than OFormer and LNO, though AROMA achieves the lowest error on the airfoil dataset.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CALM-PDE instead of LNO | regular-mesh PDE solving · 1D Burgers' ν=1e^{-3} | relative L2 error: 0.0309 -> 0.0246 | Helped |
| Use CALM-PDE instead of OFormer | regular-mesh PDE solving · 2D Navier-Stokes ν=1e^{-4} | relative L2 error: 0.0380 -> 0.0301 | Helped |
| Use CALM-PDE instead of LNO | regular-mesh PDE solving · 2D Navier-Stokes ν=1e^{-5} | relative L2 error: 0.0789 -> 0.1033 | Hurt |
| Use CALM-PDE instead of F-FNO | regular-mesh PDE solving · 3D Navier-Stokes η=ζ=1e^{-8} | relative L2 error: 0.6466 -> 0.6761 | Hurt |
| Use learnable query points instead of fixed, randomly sampled, or mesh-prior query points (ablation) | CALM-PDE · regularly and irregularly sampled PDE domains | relative L2 error | Helped |
| Add kernel modulation (ablation) | CALM-PDE · regularly and irregularly sampled PDE domains | relative L2 error | Helped |
| Add distance weighting (ablation) | CALM-PDE · PDE solving | relative L2 error | Helped |
Surprising findings
- AROMA achieves the lowest error on the airfoil dataset and CALM-PDE is outperformed by LNO and F-FNO on some regular-grid benchmarks, despite the paper's efficiency focus.
- The authors note CALM-PDE is still not efficient enough for practical 3D applications: training at 64^3 requires one A100 80GB while LNO requires four.
Limitations
- Compression into a small latent space loses fine-grained details, potentially problematic for Kolmogorov flow-type PDEs with high frequencies.
- Not efficient enough for practical applications with millions of spatial points; 3D experiments at 64^3 already need one A100 80GB GPU.
- Training is end-to-end, as a two-stage procedure was found less stable; no comparison to alternative training schemes beyond that.
Experimental setup
Datasets: PDEBench 1D Burgers' equation 2D Navier-Stokes (nu=1e-4, 1e-5) 3D compressible Navier-Stokes 2D Euler airfoil 2D incompressible Navier-Stokes cylinder
Benchmarks: PDEBench 1D Burgers 2D Navier-Stokes 3D Navier-Stokes airfoil Euler cylinder Navier-Stokes
Metrics: relative L2 error inference time training time per epoch memory consumption
Baselines: FNO Geo-FNO F-FNO OFormer PIT LNO AROMA
Builds on: deep parametric continuous convolution Position-Induced Transformer Neural ODE Fourier Neural Operator
Closest papers by meaning
- P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics (NeurIPS 2024 · similarity 0.90)
- Mesh-Free Convolution: Learned Spectral Attenuation and Transport (NeurIPS 2026 · similarity 0.90)
- Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations (ICML 2024 · similarity 0.90)
- MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation (ICML 2025 · similarity 0.90)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.89)
- AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields (NeurIPS 2024 · similarity 0.89)
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