Two orthogonal axes · 7 × 6 = 42 cells · 120 primary-source papers · through June 2026
Three architectures established operator learning as a distinct sub-field: DeepONet (branch-trunk inner product), FNO (Fourier-domain kernel), and GNO (graph message passing). Each proved a universal approximation theorem for operators and demonstrated zero-shot super-resolution on standard PDE benchmarks.
The PINO loss combines supervised data with PDE residuals at multiple resolutions; FourCastNet demonstrates FNO pretraining at climate scale. Physics-informed and foundation-model paradigms begin their rise.
Hard physics-as-architecture emerges: G-FNO embeds group equivariance, PI-GANO uses signed-distance-field embeddings for cross-geometry generalization. Theoretical rate results arrive.
Attention-based NOs compete with FNO on irregular geometry; PI-GANO formalizes "geometry as architecture"; CoDA-NO introduces codomain pretraining for foundation models.
Hybrid compositions become the SOTA for irregular geometry; IRNO formalizes FNO iteration as fixed-point solving; GAOT (NeurIPS 2025) demonstrates graph + transformer + spectral hybrids on industrial CFD.
Two converging frontiers: (1) hard physics-as-architecture — GENERIC-FNO, EqGINO, gauge-equivariant NOs compose multiple symmetry groups; (2) foundation models for physics — CompNO, Therm-FM, MoE routing. The two converge at equivariant foundation models.
The 7×6 matrix organizes primary-source Neural Operator papers along two orthogonal design axes:
The cross-product is a 7 × 6 = 42-cell matrix. As of June 2026: 19 strong cells (3+ papers), 12 single-cell (1–2 papers), 11 open (no papers), 0 special (I.4 × II.3 was promoted from Special to Single in June 2026 with GeoTransolver as new evidence; Raissi HFM and Li NMP added as additional 2025–2026 evidence). Click any cell to see its evidence, key insight, and open frontier.
| Row | Mechanism | Native cost | Native geometry |
|---|---|---|---|
| I.1 DeepONet | Branch × trunk inner product | O(Nb + Nt) | Any (sensors to points) |
| I.2 FNO | Fourier-domain global convolution | O(N log N) | Regular grids (periodic) |
| I.3 GNO | Graph message passing | O(N · |N(x)|) | Irregular / manifold |
| I.4 Transformer | Data-dependent attention kernel | O(N²) → O(M² + NM) | Any (tokenization) |
| I.5 Low-Rank | Kernel factorization / multipole | O(R · N) | Multi-scale physical |
| I.6 Mamba | Selective state-space model | O(N) | Any (with ordering) |
| I.7 Hybrid | Multi-mechanism composition | Sum of components | Component-dependent |
| Col | Paradigm | Where physics enters | Hard / soft |
|---|---|---|---|
| II.1 Pure Data | Supervised loss only | Nowhere | (no physics) |
| II.2 PINO | PDE residual loss | Loss augmentation | Soft |
| II.3 Structure | Architectural constraints | Architecture design | Hard |
| II.4 Generative | Diffusion / GAN / VAE | Output distribution | (probabilistic) |
| II.5 Iterative | Fixed-point refinement | Inference loop | (refinement) |
| II.6 Foundation | Cross-PDE pretraining | Pretraining corpus | (transfer) |
Reading the matrix: each cell aggregates primary-source papers at the intersection of an architectural mechanism (row) and a physics-integration paradigm (column). Strong cells (3+) are evidence-anchored; single cells (1–2) are nascent signals; open cells are explicitly tracked gaps in the literature — the research program’s next frontier.