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Geometry-Informed Neural Networks
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Introduces geometry-informed neural networks (GINNs), which train shape-generative neural fields with no data using geometric objectives/constraints plus a diversity constraint solved via adaptive augmented Lagrangian.
Problem
Training shape-generative models in geometry, design, and engineering when large shape datasets are unavailable; standard generative models need data and naive constrained training suffers mode collapse.
Approach
Trains conditional implicit neural shapes (SDF-like fields) purely from differentiable geometric constraints (interface, normals, design region, connectedness via persistent homology, smoothness via curvature) and a minimal-aggregation diversity constraint based on boundary-based shape distance, optimized with adaptive augmented Lagrangian.
Key results
- Diversity constraint prevents mode collapse in the obstacle problem: delta_mean rises from 0.08-0.12 range without it to 0.15 with diversity loss in the single-shape ablation table (ablation tables show diversity ablated to 0.12, full model 0.15).
- Jet engine bracket multi-shape ablation: removing the connectedness constraint raises b0 to 25.11 and reduces connected interfaces to 0.17, showing its necessity.
- Runtime of about 10K iterations (30 minutes) for a single bracket shape and about 50K iterations (5 hours) for the diverse model on a single GPU.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add diversity constraint (ablation) | GINN · Obstacle (multiple shapes) | δ_mean: 0.12 -> 0.15 | Hurt |
| add connectedness constraint (ablation) | GINN · Obstacle (multiple shapes) | δ_mean: 1.45 -> 0.15 | Hurt |
| add interface constraint (ablation) | GINN · Obstacle (multiple shapes) | b₀(Ω): 0.00 -> 1.11 | Mixed |
| add design-region constraint (ablation) | GINN · Obstacle (multiple shapes) | vol(Ω \ E) / vol(X \ E): 0.53 -> 0.00 | Helped |
| add diversity constraint (ablation) | GINN · Wheel (multiple shapes) | δ_mean: 0.03 -> 0.31 | Helped |
| add design-region constraint (ablation) | GINN · Wheel (multiple shapes) | b₀(Ω): 6.89 -> 1.11 | Helped |
| add connectedness constraint (ablation) | GINN · Wheel (multiple shapes) | b₀(Ω): 1.22 -> 1.11 | Helped |
| add rotational-symmetry constraint (ablation) | GINN · Wheel (multiple shapes) | δ_mean: 0.54 -> 0.31 | Hurt |
| add eikonal constraint (ablation) | GINN · Wheel (multiple shapes) | δ_mean: 0.03 -> 0.31 | Helped |
| add diversity constraint (ablation) | GINN · Jet engine bracket (multiple shapes) | δ_mean: 0.05 -> 0.14 | Helped |
| add connectedness constraint (ablation) | GINN · Jet engine bracket (multiple shapes) | b₀(Ω): 25.11 -> 1.00 | Helped |
| add surface-strain constraint (ablation) | GINN · Jet engine bracket (multiple shapes) | E_strain(Ω): 263 -> 410 | Hurt |
Surprising findings
- Simple generative GINN models without a diversity term are prone to mode collapse, and the diversity constraint also induces an organized, interpretable latent space.
- The first PINN reported to produce 2D Turing patterns in a data-free setting, using stationary reaction-diffusion equations without initial conditions.
Limitations
- Did not compare against established topology-optimization methods since this required a PDE solver integration, deferred to future work.
- Diversity defined on finite batches and the distance is not a metric; the latent structure depends on the exact diversity setup and over-specifying the threshold impedes convergence.
- Computational cost is high: persistent homology dominates runtime and several losses (diversity, smoothness) increase iteration counts substantially.
Experimental setup
Benchmarks: obstacle wheel jet engine bracket Plateau's problem parabolic mirror Turing patterns (reaction-diffusion)
Metrics: Betti number b0 (connectedness) normalized volume outside design region chamfer distance to interface (CD1) diversity delta_mean surface strain share of connected interfaces
Baselines: softplus-MLP SIREN WIRE
Builds on: physics-informed neural networks adaptive augmented Lagrangian method SIREN WIRE Boltzmann generators
Closest papers by meaning
- Geometry-Aware Neural Optimizer for Shape Optimization and Inversion (ICML 2026 · similarity 0.86)
- Improving Neural Network Surface Processing with Principal Curvatures (NeurIPS 2024 · similarity 0.85)
- Geometry Awakening: Cross-Geometry Learning Exhibits Superiority over Individual Structures (NeurIPS 2024 · similarity 0.85)
- Enhancing Implicit Shape Generators Using Topological Regularizations (ICML 2024 · similarity 0.85)
- DTGBrepGen: A Novel B-rep Generative Model through Decoupling Topology and Geometry (CVPR 2025 · similarity 0.85)
- PoNQ: a Neural QEM-based Mesh Representation (CVPR 2024 · similarity 0.85)
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