A quadrilateral block decomposition of a planar domain is judged by whether it is complete, whether its elements are well shaped, and how many of its vertices are irregular. The last has a provable floor: the discrete Gauss-Bonnet identity enforces a lower bound on the total vertex irregularity of any all-quadrilateral mesh of a given domain purely based on its topology and corner angles. We train a reinforcement learning agent to build decompositions that reach this bound, which we call par. It acts directly on the mesh's half-edge data structure through local edits, with a policy network whose convolutions follow the mesh's own connectivity, so it applies unchanged to domains larger than any seen in training. The reward targets the floor directly, and it is sparse: random play reaches it on no domain with more than eight sides. We overcome this exploration barrier via behaviour cloning on optimal meshes that are trivial to construct, walked backward into demonstrations, before training it with PPO. On 96 held-out domains the agent produces an all-quadrilateral mesh on every one, a usable one on 95.7 on average, and a provably optimal one on 90; Gmsh's strongest configuration at the same element count completes 51, is usable on 38 and optimal on none, and even at three to fourteen times the elements never produces a more regular mesh. On 64 domains twice the training size the agent completes all, is usable on 62, and keeps a median excess over par below one against Gmsh's 39 at the same element count.
Figures & tables
Figure 1: Our agent and Gmsh on the same three domains. Top: an in-distribution domain (chamfered, with a hole, 18 corners). Middle: a larger-boundary domain ( 38 corners), beyond any seen in training. Bottom: a curved domain with holes ( 33 corners); the agent was never trained on a curved boundary, and quality is given against the arc tangents and on chords (Section 6.4 ). Gmsh blossom is asked for our element count and returns the closest it can reach; quasi-structured runs at its natural size. Every irregular vertex is marked. Our top mesh has none, which the Gauss–Bonnet bound (Section 2 ) certifies as optimal; the middle one has one, against 32 and 16 for Gmsh, and the curved one 15 , against 64 and 35 . Domains are chosen by the rule of Appendix M , which shows one per family.
Figure 2: (a) A mesh is stored as a doubly connected edge list: every edge carries two oppositely oriented half-edges, and all connectivity is expressed through next and prev , which walk a face loop counter-clockwise, and twin , which crosses to the adjoining face. A half-edge on ∂Ω has no twin, and the network is told so by a sentinel distinct from the one marking a neighbour that merely fell outside the window. (b) From a half-edge h with source u , three actions insert a chord to the vertex j+2 positions ahead in the face loop, cutting off a sub-face of j+3 sides; the fourth splits h ’s own edge with a new vertex. Chords leaving a 2 -gon on either side are masked out.
our win rate
all-quad
usable
excess
at par
elements
× ours
quality
regularity
96 in-distribution domains ( 8 – 24 corners)
Gmsh blossom
51
38
9
0
19
1.2
90%
100%
Gmsh frontal-quad
43
29
8
1
16
1.0
90%
100%
Gmsh quasi-structured
96
96
7.5
2
51
3.2
67%
100%
Gmsh blossom-full
96
96
26
0
220
14.0
37%
100%
Table 1: Top: the 96 in-distribution domains, mean over six evaluation seeds (per family: Table 7 ). Bottom: the 64 larger-boundary domains, four evaluation seeds; its last rows remove parts of the evaluation-time procedure (cumulatively) and add attempts. Columns and win rates as defined in Section 6.1 ; each Gmsh configuration is asked for our element count and returns the closest it can reach. Bold: best in column within a block (elements: among methods complete on every domain).
usable
quality win
all-quad
excess
at par
elem.
× ours
reg. win
tangent
chord
tangent
chord
Gmsh blossom
62
20
0
60
1.1
100%
7
54
94%
54%
Gmsh frontal-quad
55
22
0
58
1.1
100%
2
42
100%
78%
Gmsh quasi-structured
96
10
0
83
1.5
100%
6
64
95%
67%
Gmsh blossom-full
93
32
0
235
4.4
100%
25
79
74%
47%
Ours
96
2
37
54
1.0
95
88
Table 2: Curved boundaries: the released agent without retraining, one evaluation pass on 96 domains with 8 – 24 corners. Left: columns that do not depend on how quality is measured, as in Table 1 (at par with q≥0.1 on the tangents; on chords quasi-structured reaches par on 2 ). Right: usable count and our win rate on quality, with element quality judged against the arc tangents or chords. Ours: with the repair search; the row below it is the five attempts and doubled budget of Section 6.1 without any repair. Bold: best in column (elements: among methods complete on every domain).
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Certified instances, one per family, each the first accepted at a fixed seed. Columns: the constructed optimal mesh; the backward walk a third of the way through; the single face it ends on, which is the start state; and the training instance, the transformed outline with the recorded moves replayed on it and accepted as solved. The last three rows are drawn with rotation and scale undone. Holed outlines keep one edge joining the hole to the outer boundary, so that the start state is a single face. Shaded: faces that are not quadrilaterals.
set
sub-problems
par 0
1
2
3
≥4
par >0
episodes
96 in-distribution
2,195
60.7%
33.8%
4.6%
0.8%
0.1%
39.3%
75 of 96
64 larger
11,717
26.7%
59.0%
10.9%
3.0%
0.4%
73.3%
64 of 64
Appendix
Table 3: Bounds of the sub-problems met in the released agent’s greedy attempt on the par-zero evaluation domains; the last column counts episodes that meet at least one sub-problem with par above zero.
par 1 ( 11 )
2 ( 17 )
3 ( 17 )
4 ( 3 )
total
all-quad
Ours
11
16
17
3
47
48
Gmsh quasi-structured
2
6
0
0
8
48
Gmsh packing
0
3
4
0
7
8
Gmsh frontal-quad
1
0
0
0
1
27
Gmsh blossom
0
0
0
0
0
30
Gmsh blossom-full
0
0
0
0
0
9
Appendix
Table 4: 48 held-out domains with par above zero: domains at par, by par. Ours: best of five attempts, ranked by completion and then closeness to par, at the default move budget. Gmsh: at the agent’s element count where it can reach it.
Figure 4: The policy. The trunk is per half-edge: convolved features pass through a shared per-slot linear layer to the action head. Pooling is a branch — it collapses the half-edge axis, and the global vector and progress scalar, which never enter message passing, join there. The context vector it produces is one vector per state: the critic’s only input, and, on the actor side, an identical shift added to every half-edge before a shared linear head. It can therefore re-weight the four action types but cannot by itself favour one half-edge over another — selecting where to act rests on the convolution and on the rule that decides which half-edges are in the window at all.
observation window nT
64 half-edges
edge-length feature
divided by the window’s median
global scalars
per face, seven entries
actions per half-edge
4 : chords to +2 , +3 , +4 ; vertex insertion
default move budget Tmax
max(12,⌈3∣H0∣⌉)
smoothing per move
5 Laplacian sweeps, corners pinned
potential weights wv , wq , q⋆
1 , 8 , 0.4
Appendix
Table 5: The released configuration.
our win rate
all-quad
usable
excess
at par
elements
× ours
quality
regularity
Gmsh blossom
51
38
9
0
19
1.2
90%
100%
Gmsh frontal-quad
43
29
8
1
16
1.0
90%
100%
Gmsh packing
8
6
5
0
15
0.9
98%
100%
Gmsh simple-recombine
2
1
11
0
14
0.9
100%
100%
Gmsh subdivide
96
43
49
0
85
5.4
97%
100%
Appendix
Table 6: Seven Gmsh configurations on the 96 in-distribution domains, each asked for the agent’s element count where it can produce one; the elements column gives the mean over its all-quadrilateral meshes and its multiple of our 15.7 . On packing and simple-recombine nearly every win is decided by completion. Bold: best in column; for elements, among methods complete on every domain (packing and simple-recombine average over 8 and 2 meshes). Our win rate: fraction of decided domains on which our mesh is better on quality and on regularity, under the rule of Section 6.1 .
Figure 5: Per-mesh minimum quality and excess over par, one mesh per domain: the agent in one evaluation pass (rollout seed 0 ) and Gmsh asked for that pass’s element count on each domain. Quality panels: domains per bin, hatched bar not all-quadrilateral. Excess panels: share of the method’s all-quadrilateral meshes, with the number at zero quoted. The counts are this pass’s; the tables report the mean over evaluation seeds.
family
method
all-quad
usable
excess
at par
elements
chamfered
Ours
24.0
24.0
0
20.7
14
Gmsh blossom
14
12
8.5
0
19
Gmsh quasi-structured
24
24
6.5
0
54
rectilinear
Ours
24.0
24.0
0
23.8
10
Gmsh blossom
16
10
8
0
15
Gmsh quasi-structured
24
24
5
1
38
Appendix
Table 7: The in-distribution block of Table 1 by family: best of five attempts under the procedure, mean over six evaluation seeds. Columns in the order a mesher is judged: all-quadrilateral, usable ( q≥0.3 ), median excess over par, and at par (the certified optimum). Gmsh blossom is given the agent’s element count on each domain; quasi-structured cannot go that coarse and runs at its natural size, 3.2× the agent’s. Bold: best in column over all 96 domains.
set
column
mean
sd
range
96 in-distribution, 6 seeds
all-quad
96.0
0.0
96 – 96
usable
95.7
0.5
95 – 96
excess
0
0.0
0 – 0
at par
90.2
1.3
88 – 92
64 larger, 4 seeds
all-quad
64.0
0.0
64 – 64
usable
62.0
1.0
61 – 63
Appendix
Table 8: The released agent under the procedure of Section 6.1 : mean, standard deviation across evaluation seeds, and range.
extractor
params
at par
DCEL, 4 layers
299 k
94.5
Transf., 4 layers
300 k
51.5
Appendix
Table 9: Left: domains at par of the 96 in-distribution domains, best of five, at a matched 2 M steps under an earlier configuration of the recipe, mean of two seeds. Right: PPO from a fresh network, cloning only, and cloning then PPO, at 1 M PPO steps, seed 1 , best of five at the default move budget and without repair.
usable
quality win
suite
method
all-quad
excess
at par
elem.
× ours
tangent
chord
tangent
chord
8 – 24
Gmsh blossom
35
18
0
60
1.1
3
31
94%
50%
Gmsh frontal-quad
30
18
0
58
1.1
2
23
100%
77%
Gmsh quasi-structured
48
9
0
81
1.6
3
33
94%
71%
Gmsh blossom-full
48
31
0
249
4.8
13
44
73%
46%
Ours
48
2
21
52
1.0
47
45
Appendix
Table 10: Table 2 by suite, with the larger domains; every suite is curved. Columns, rows and win rates as there; our regularity win rate is 100% against every configuration in every suite and is omitted. Bold: best in column within a suite. The holed larger suite has 16 domains, the others 48 .
Figure 6: One domain per in-distribution family ( 8 – 24 corners).
Figure 7: Larger-boundary domains ( 25 – 50 corners), beyond any seen in training: one per family.
Figure 8: Curved domains, one per suite, meshed by the released agent, trained only on straight-sided domains, with the repair search, and by Gmsh as in Figure 1 . A boundary edge that lies on an arc is drawn along the arc; ours has a boundary node every 45∘ of arc. Each panel gives the minimum element quality judged against the arc tangents and on the straight sides (Section 6.4 ); the hatching uses the tangent.
Generating high-quality meshes for arbitrary geometries remains a fundamental bottleneck in computational engineering, often demanding heuristic tuning and semi-manual workflows. In this paper, we introduce Dmsh, a first fully automated reinforcement learning pipeline that unifies geometric decomposition and quadrilateral mesh generation within a single learning-based framework. Dmsh decomposes the problem through three coordinated agents handling topology simplification, geometric regularization, and mesh generation. The meshing process is formulated as a Markov Decision Process and solved using a parametric Soft Actor-Critic architecture with decoupled critics, enabling efficient exploration of a hybrid discrete-continuous action space. A curriculum learning strategy ensures scalability from simple domains to highly complex geometries, suppressing seed variance. By design, the recursive decomposition enables parallel meshing of subregions, yielding globally conforming all-quadrilateral meshes without post hoc correction. Across a wide range of benchmarks, Dmsh consistently outperforms existing methods in automation, robustness, and mesh quality, establishing a new paradigm for learning-based mesh generation.
Reinforcement learning in large or sparse-reward environments suffers from slow temporal-difference reward propagation, as value information spreads only locally across the state space. We propose Mesh-RL, a spatial domain-decomposition framework inspired by the finite element method and domain decomposition theory, which partitions the environment into overlapping subgrids and enforces boundary-consistent temporal-difference updates. Such an approach enables localized learning while ensuring globally coherent value propagation. Unlike hierarchical or model-based approaches, Mesh-RL accelerates long-range credit assignment without modifying the reward function, Bellman operator, or introducing explicit planning mechanisms. We evaluate Mesh-RL on hazard-dense grid-world environments with varying geometries and mesh resolutions. Across Q-learning, SARSA, and Dyna-Q, Mesh-RL consistently improves convergence speed, cumulative reward, and learning stability. Higher mesh resolutions sustain exploration, prevent premature convergence, and substantially accelerate value propagation to distant states. While Dyna-Q already benefits from internal planning, it still achieves additional gains under structured decomposition. Overall, Mesh-RL introduces a principled spatial domain-decomposition mechanism for accelerating temporal-difference learning. Our framework bridges finite element method-inspired boundary-consistency techniques from scientific computing with reinforcement learning to improve sample efficiency in sparse-reward environments. We will release source code of the study.
Reinforcement learning (RL) has demonstrated remarkable success in text and image generation, yet its potential in 3D generation remains largely unexplored. Existing attempts typically rely on offline direct preference optimization (DPO) method, which suffers from low training efficiency and limited generalization. In this work, we aim to enhance both the training efficiency and generation quality of RL in 3D mesh generation. Specifically, (1) we design the first asynchronous online RL framework tailored for 3D mesh generation post-training efficiency improvement, which is 3.75× faster than synchronous RL. (2) We propose Advantage-guided Ranking Preference Optimization (ARPO), a novel RL algorithm that achieves a better trade-off between training efficiency and generalization than current RL algorithms designed for 3D mesh generation, such as DPO and group relative policy optimization (GRPO). (3) Based on asynchronous ARPO, we propose Mesh-Pro, which additionally introduces a novel diagonal-aware mixed triangular-quadrilateral tokenization for mesh representation and a ray-based reward for geometric integrity. Mesh-Pro achieves state-of-the-art performance on artistic and dense meshes.
Zhen Zhou, Jian Liu, Biwen Lei +10
Institute of Automation, Chinese Academy of Sciences · Tencent Hunyuan · Hong Kong University of Science and Technology