# RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains

arXiv: https://arxiv.org/abs/2501.19205 · HTML: https://mlsanity.com/paper/2501.19205/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2025.

## Built on by (most cited first, 23 of 23)

- [Rectified Flows for Fast Multiscale Fluid Flow Modeling](https://mlsanity.com/paper/2506.03111/) (2025)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Imposing Boundary Conditions on Neural Operators via Learned Function Extensions](https://mlsanity.com/paper/2602.04923/) (2026)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)
- [Graph In-Context Operator Networks for Generalizable Spatiotemporal Prediction](https://mlsanity.com/paper/2603.12725/) (2026)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation](https://mlsanity.com/paper/2605.24041/) (2026)
- [Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training](https://mlsanity.com/paper/2606.09949/) (2026)
- [Neural Shape Operator Surrogates -- Expression Rate Bounds](https://mlsanity.com/paper/2604.18012/) (2026)
- [GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems](https://mlsanity.com/paper/2607.01128/) (2026)
- [From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning](https://mlsanity.com/paper/2608.06894/) (2026)
- [Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulations](https://mlsanity.com/paper/2601.12614/) (2026)
- [Loss Landscape Geometry of Partial Differential Equation Emulators: Or, Symmetry Learning via Gradient Alignment](https://mlsanity.com/paper/2601.20172/) (2026)
- [AeTHERON: Autoregressive Topology-aware Heterogeneous Graph Operator Network for Fluid-Structure Interaction](https://mlsanity.com/paper/2604.13369/) (2026)
- [IKNO: Infinite-order Kernel Neural Operators](https://mlsanity.com/paper/2605.22182/) (2026)
- [Temperature Field Reconstruction of Tungsten Monoblock Divertor on EAST using Physics-aware Neural Operator Transformer](https://mlsanity.com/paper/2606.31574/) (2026)
- [FLARE++: Low-rank attention with dynamic attention routing](https://mlsanity.com/paper/2608.11519/) (2026)
- [STCO: Conditional Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2608.20477/) (2026)
- [Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators](https://mlsanity.com/paper/2609.05860/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)
- [Does Transolver really need a Transformer?](https://mlsanity.com/paper/2609.32525/) (2026)
- [Geometry-physics confounding impairs PDE learning across varying domains](https://mlsanity.com/paper/2609.38623/) (2026)
- [Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers](https://mlsanity.com/paper/2610.03363/) (2026)

## Abstract (arXiv)

Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions both in space and in time. Our code is publicly available at github.com/camlab-ethz/rigno.