Research map / Label-efficient and robust learning / Zero-/few-shot and meta-learning

UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss

NeurIPS 2025 · Label-efficient and robust learning › Zero-/few-shot and meta-learning · approach: Zero-shot robustness and transfer · 0 citations (OpenAlex)

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 UGM2N, an unsupervised, patch-based graph-transformer mesh movement network trained with a variance-based M-Uniform loss, achieving zero-shot generalization across PDEs and mesh geometries without pre-adapted meshes.

Problem

Mesh movement (r-adaptation) for PDE solvers is computationally expensive with Monge-Ampère methods, and existing supervised learning approaches (M2N, UM2N) need pre-adapted meshes and generalize poorly across PDEs and mesh topologies.

Approach

Process local node patches (a node and its first-order neighbors, normalized to a unit square) with a graph transformer and train unsupervised using an M-Uniform loss that penalizes variance of per-element equidistribution quantities within each patch, so no pre-adapted mesh labels are needed.

Key results

• On the Poisson benchmark, UGM2N achieved the highest ER in five of seven cases, e.g. 9.00% ER on the sum-of-Gaussians solution versus -8.64% for MA and -5.59% for UM2N.
• On Helmholtz, UGM2N secured the top ER in four of five cases (e.g., 16.98% vs 13.50% for MA on cos(4πy)cos(2πx)).
• On 100 non-convex meshes, UGM2N and UM2N produced 0% tangling (100/100 valid meshes) while MA produced valid meshes in only 41 cases.

Measured effects reported in the paper

Surprising findings

• M2N achieved strongly negative error reduction on many Poisson and Helmholtz cases (e.g., -351.63% on one Helmholtz case), indicating supervised MSE-trained adaptation can severely degrade accuracy on unseen PDEs.
• Clipping boundary nodes back to the domain boundary improved ER on several Poisson cases by over 12 percentage points, despite the default fixed-boundary strategy.

Limitations

• I/O overhead between GPU and CPU for Delaunay interpolation dominates runtime and limits scalability to around 40,000 elements on a single GPU.
• Boundary nodes are kept fixed, and naive clip projection is only a partial remedy; the paper gives no convergence guarantee for the iterative Jacobi-style updates.

Experimental setup

Datasets: four analytical flow fields on a triangular mesh with random node perturbations

Benchmarks: Poisson equation (7 solutions) Helmholtz equation (5 solutions) Burgers equation (2 initial conditions) NACA0012 airfoil cylinder flow wave equation on circular domain moving cylinder supersonic flow over wedge non-convex mesh tangling test 1,000 random polygonal domains

Metrics: error reduction (ER) tangling ratio (TR) MAE of Cp and C_D L_var on adapted mesh mesh movement time

Baselines: MA method M2N UM2N

Builds on: UM2N M2N Monge-Ampère mesh adaptation Graph Attention Network PINNs

Closest papers by meaning

• Towards Universal Mesh Movement Networks (NeurIPS 2024 · similarity 0.92)
• Better Neural PDE Solvers Through Data-Free Mesh Movers (ICLR 2024 · similarity 0.88)
• PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.87)
• PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations (ICLR 2025 · similarity 0.86)
• A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds (NeurIPS 2026 · similarity 0.86)
• PIORF: Physics-Informed Ollivier-Ricci Flow for Long–Range Interactions in Mesh Graph Neural Networks (ICLR 2025 · similarity 0.86)