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Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations
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Proposes reference neural operators (RNO) that learn the material-derivative-style change in solution caused by a geometric deformation from a reference solution, achieving strong accuracy with small datasets.
Problem
Learning neural operators that map geometries to PDE solutions requires large geometry-solution datasets, which are prohibitive for expensive industrial simulations such as engineering design optimization.
Approach
Instead of learning geometry-to-solution maps, learn the difference between the queried solution and the pushforward of a reference solution under a smooth deformation, using a distance-aware cross attention (with a linear-complexity random-feature variant) to combine reference solution, deformation, and geometry inputs.
Key results
- On NS2d-360, RNO achieves u error 3.9e-2 vs 1.6e-1 for GNOT and 4.6e-1 for Geo-FNO, up to 80% error reduction across datasets.
- On Inductor2d-320, RNO reaches A_z error 4.3e-3 vs 2.5e-2 for the best baseline R-GNOT.
- On NS2d-1440, RNO reduces u/v/p errors to 1.9e-2/5.1e-2/2.3e-2 versus 4.3e-2/1.1e-1/4.4e-2 for R-GNOT.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RNO to predict the solution from a reference solution and geometry deformation | RNO vs. strongest listed baseline · NS2d-360, NS2d-sq-360, Inductor2d-320, Heatsink3d-80 | relative L2 error: NS2d-360 u: 1.0e-1 -> 3.9e-2; Inductor2d-320 Az: 2.5e-2 -> 4.3e-3 | Helped |
| Predict δu and add it to the pushed-forward reference solution instead of predicting u_q directly (ablation) | RNO · NS2d, NS2d-sq, Inductor2d | relative L2 error: 7.4e-2 -> 6.3e-2; 1.2e-1 -> 1.1e-1; 3.6e-2 -> 1.5e-2 | Helped |
| Add DACA layers (ablation) | RNO · NS2d, NS2d-sq, Inductor2d | relative L2 error: 9.9e-2 -> 6.3e-2; 2e-1 -> 1.1e-1; 3.8e-2 -> 1.5e-2 | Helped |
| Replace full DACA with linear DACA using random feature mapping (ablation) | RNO-L · NS2d-360 and NS2d-1440 | relative L2 error: NS2d-360 u: 3.9e-2 -> 4.2e-2; NS2d-1440 u: 1.9e-2 -> 2.5e-2; NS2d-360 v: 1.1e-1 -> 1.1e-1 | Mixed |
| Increase the number of RNO layers beyond 3 (ablation) | RNO · RNO experiments | relative L2 error | No clear effect |
| Increase DACA γ (ablation) | RNO · NS2d | mean relative L2 error | Hurt |
| Increase DACA γ (ablation) | RNO · Inductor2d | mean relative L2 error | Helped |
| Use the reference pushforward alone, without predicting δu (ablation) | RNO · NS2d, NS2d-sq, Inductor2d, Heatsink3d | mean relative L2 error: 6.3e-2 -> 1.2e-1; 1.1e-1 -> 2.5e-1; 1.5e-2 -> 7.3e-2; 8.6e-2 -> 1.6e-1 | Hurt |
| Increase the training dataset size (ablation) | RNO · NS2d | relative L2 error | Helped |
Surprising findings
- Error increases with geometric distance for NS2d but decreases with gamma (distance weighting disabled is best) for Inductor2d, reflecting the slow-decay kernel of elliptic Maxwell equations.
- On Airfoil-RANS, RNO did not win on every component: R-GNOT was best on u, v and nu_t, with RNO best only on pressure.
Limitations
- Requires a smooth deformation between reference and query domains; topology changes such as adding or removing holes are not supported.
- Constructing the deformation relies on a non-smooth one-to-one boundary matching step and needs a nearest-neighbor reference selection.
- Performance depends on the distance-weight hyperparameter gamma, which is problem dependent.
Experimental setup
Datasets: NS2d NS2d-sq Inductor2d Heatsink3d Airfoil-Euler Airfoil-RANS
Benchmarks: NS2d-360 NS2d-sq-360 Inductor2d-320 Heatsink3d-80 NS2d-1440 Airfoil-Euler Airfoil-RANS
Metrics: relative L2 error
Baselines: GNOT R-GNOT Geo-FNO R-FNO-i R-MIONet CORAL NU-FNO pushforward of reference solution
Builds on: Geo-FNO GNOT Fourier Neural Operator DeepONet random feature mapping
Closest papers by meaning
- Adaptive Coordinate Transforms for Neural Operators (NeurIPS 2026 · similarity 0.89)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.88)
- BENO: Boundary-embedded Neural Operators for Elliptic PDEs (ICLR 2024 · similarity 0.88)
- Accurate Differential Operators for Hybrid Neural Fields (CVPR 2025 · similarity 0.87)
- Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws (ICML 2024 · similarity 0.87)
- Neural Green’s Functions (NeurIPS 2025 · similarity 0.87)
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