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Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs
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Proposes the inducing-point operator transformer (IPOT), an attention-based encoder-processor-decoder that compresses arbitrary input functions into a fixed set of learnable latent inducing points, processes them, and decodes at arbitrary output queries, scaling linearly in input/output size.
Problem
Learning PDE solution operators that flexibly handle irregular, arbitrary input/output discretizations while scaling to large discretizations and long-horizon forecasting.
Approach
Use cross-attention with a small number of learnable inducing-point queries as encoder, a latent self-attention processor independent of input/output discretization, and cross-attention decoder at arbitrary query points, so cost is linear in input/output size and processor depth is decoupled from data size.
Key results
- On ERA5 T2m forecasting, IPOT reaches 6.64e-3 relative L2 error vs 7.25e-3 for FFNO and 1.21e-2 for FNO, with 0.51M parameters.
- On Navier-Stokes (nu=1e-3), IPOT achieves 8.85e-3 error with 0.12M parameters and 21.05 s/epoch, versus 1.04e-2 for OFormer.
- On the plasticity benchmark IPOT achieves 3.25e-3 error, better than FFNO's 4.70e-3, while FFNO wins on Darcy flow (7.70e-3 vs 1.73e-2 for IPOT).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IPOT instead of FFNO | Darcy Flow · Darcy Flow | relative L2 error: 7.70e--3 -> 1.73e--2 | Hurt |
| Use IPOT instead of OFormer | Navier Stokes · Navier Stokes | relative L2 error: 1.04e--2 -> 8.85e--3 | Helped |
| Use IPOT instead of FFNO | Airfoil · Airfoil | relative L2 error: 7.80e--3 -> 8.79e--3 | Hurt |
| Use IPOT instead of OFormer | Elasticity · Elasticity | relative L2 error: 1.83e--2 -> 1.56e--2 | Helped |
| Use IPOT instead of FFNO | Plasticity · Plasticity | relative L2 error: 4.70e--3 -> 3.25e--3 | Helped |
| Use IPOT instead of FFNO | ERA5 · ERA5 | relative L2 error: 7.25e--3 -> 6.64e--3 | Helped |
| Use 64 inducing points rather than the 512-point variant (ablation) | IPOT · ERA5 | relative L2 error: 6.44e--3 -> 1.45e--2 | Hurt |
| Use 128 inducing points rather than the 512-point variant (ablation) | IPOT · ERA5 | relative L2 error: 6.44e--3 -> 1.30e--2 | Hurt |
| Increase inducing points from 256 to 512 (ablation) | IPOT · ERA5 | relative L2 error: 6.87e--3 -> 6.44e--3 | Helped |
| Remove inducing points and use observational self-attention (ablation) | IPOT · ERA5 | runtime (seconds) | Hurt |
Surprising findings
- On Darcy flow and Airfoil, FFNO attains lower error than IPOT (7.70e-3 and 7.80e-3 vs 1.73e-2 and 8.79e-3), so IPOT is not uniformly best.
- Removing inducing points (full self-attention over observations) did not converge or ran over 100 s/epoch, showing quadratic attention is impractical at this scale.
Limitations
- Results for baselines on standard benchmarks are copied from prior literature, and IPOT is not best on every benchmark (e.g., Darcy flow, Airfoil).
- Small number of inducing points (64) degrades performance noticeably (error 1.45e-2 on ERA5), so the bottleneck size requires tuning.
Experimental setup
Datasets: Burgers Darcy flow Navier-Stokes Airfoil Elasticity Plasticity Spherical shallow water ERA5
Benchmarks: Burgers Darcy flow Navier-Stokes Airfoil Elasticity Plasticity ERA5 Spherical shallow water
Metrics: relative L2 error parameter count runtime per epoch CUDA memory
Baselines: DeepONet MeshGraphNet FNO FFNO OFormer DINo Perceiver IO
Builds on: Perceiver Fourier neural operator OFormer DeepONet Set Transformer
Closest papers by meaning
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.86)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.86)
- Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning (ICML 2024 · similarity 0.85)
- Efficient Generative Transformer Operators for Million-Point PDEs (NeurIPS 2026 · similarity 0.85)
- Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations (ICML 2024 · similarity 0.85)
- Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains (NeurIPS 2025 · similarity 0.84)
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