Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Adaptive Coordinate Transforms for Neural Operators
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes the Adaptive Coordinate Transform (ACT) block, a plug-and-play learned coordinate-resampling module interleaved layer-wise in neural operators (FNO, CNextU, Transolver) to reduce spatial misalignment and counter smoothness bias on PDE benchmarks.
Problem
Neural operators are built on fixed Eulerian coordinates, which misalign with evolving physical structures, producing non-local operator mappings and a smoothness bias near sharp transitions.
Approach
Each layer predicts bounded residual coordinate offsets from the current feature map and resamples features via differentiable grid sampling with periodic remapping, inserted after every operator block so the network adaptively realigns coordinates throughout depth.
Key results
- ACT reduces NRMSE on all 18 backbone-dataset pairs; average relative error reduction 76.4% for Transolver, 38.4% for FNO, 30.8% for CNextU.
- On Navier-Stokes with FNO backbone, NRMSE falls from 2.22% (vanilla) and 1.80% (2x params) to 1.15% with a final ACT only and 0.62% with layer-wise ACT.
- Transolver on Navier-Stokes drops from 32.29% to 1.35% NRMSE with ACT (95.8% reduction).
Surprising findings
- A single terminal ACT block on FNO already gives a large gain (2.22% to 1.15%), exceeding a model with twice the parameters (1.80%).
- Transolver baseline's NRMSE on Navier-Stokes (32.29%) is extremely high relative to its ACT variant, suggesting possibly unstable vanilla training.
Limitations
- Benefit appears smaller when dynamics are well represented in fixed coordinates or lack sharp transitions (stated).
- Mechanistic claims are supported empirically, not by formal theory.
- Restricted to structured grids; irregular meshes and complex geometries not handled.
Experimental setup
Datasets: Navier-Stokes (FNO benchmark) PDEBench diffusion-reaction PDEBench shallow water The Well active matter The Well turbulent radiative layer 2D (TRL_2D) The Well MHD 64
Benchmarks: Navier-Stokes Diff_Reaction Shallow_Water Active_Matter TRL_2D MHD_64
Metrics: NRMSE
Baselines: FNO CNextU Transolver++
Builds on: Fourier Neural Operator Transolver++ ConvNeXt-UNet deformable attention deformable convolution
Closest papers by meaning
- Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations (ICML 2024 · similarity 0.89)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.88)
- Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains (NeurIPS 2025 · similarity 0.88)
- Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning (ICML 2024 · similarity 0.88)
- Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning (NeurIPS 2024 · similarity 0.87)
- Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.