Geometry-Aware Neural Optimizer (GANO)

• The paper introduces a geometry-aware framework that optimizes a latent geometry code using an implicit SDF auto-decoder and a geometry-injected field surrogate for PDE-driven shape inversion and design.
• It leverages denoising-based latent regularization, null-space projection for part protection, and remeshing-free projection to preserve geometric fidelity and ensure smooth updates.
• Empirical evaluations across Helmholtz, airfoil, and vehicle benchmarks show improved accuracy and performance compared to traditional mesh-based optimization pipelines.

Searching arXiv for the primary GANO paper and closely related geometry-aware optimization/operator papers. Searching arXiv for "Geometry-Aware Neural Optimizer for Shape Optimization and Inversion" (Sun et al., 6 May 2026). Geometry-Aware Neural Optimizer (GANO) denotes an end-to-end differentiable framework for shape optimization and shape inversion in PDE-governed systems, where the optimization variable is a geometry latent code rather than a mesh, CAD parameter vector, or scalar objective surrogate. In its canonical formulation, the target problem is

$$\Gamma^{\star}=\arg\min_{\Gamma}\; \mathcal{J}\big(\mathbf{u}(\cdot;\Gamma)\big),$$

with $\Gamma$ the geometry, $\Omega(\Gamma)$ the induced domain, $\mathbf{u}(\cdot;\Gamma)$ the PDE solution field, and $\mathcal{J}$ either a design objective or an observation-matching objective. GANO unifies geometry representation, field-level prediction, and automated optimization or inversion in a single latent-space loop; it encodes shapes with an auto-decoder, stabilizes latent updates via a denoising mechanism, conditions a surrogate directly on geometry latent code, supports part-wise control through null-space projection, and uses remeshing-free projection to accelerate geometry processing (Sun et al., 6 May 2026).

1. Problem class and conceptual scope

GANO is designed for PDE-governed settings in which geometry is itself the control variable. The motivating applications are obstacle reconstruction in Helmholtz scattering, airfoil design under aerodynamic criteria, and vehicle drag reduction. The central difficulty is not only that forward simulation is expensive, but that the objective is defined on fields or derived forces while geometry is often represented by non-differentiable objects or manually designed parameters. Classical pipelines therefore alternate between forward simulation, geometry editing, and remeshing, which the framework explicitly treats as costly and brittle (Sun et al., 6 May 2026).

This distinguishes GANO from several adjacent notions of “geometry-aware” optimization. It is not primarily a parameter-manifold optimizer for constrained neural weights, as in Stiefel-aware federated SPDnet training (Pautrel et al., 24 Apr 2026), nor a Fisher-structured optimizer for matrix-valued layers such as FISMO (Xu et al., 29 Jan 2026). It is also not a forward neural operator in the sense of PI-GANO, which generalizes across PDE parameters and domain geometries but is formulated for operator learning rather than latent geometry optimization (Zhong et al., 2024). In GANO, geometry is the optimized object, and the differentiable loop is built to propagate objective gradients back to a shape code rather than merely to network parameters (Sun et al., 6 May 2026).

A recurrent misconception is that a neural surrogate alone “closes the loop” for shape design. GANO is explicit that forward surrogates do not solve geometry optimization by themselves because gradients from objectives to geometry are often unavailable, especially when geometry updates leave the differentiable path and pass through remeshing or manual geometry processing. A second misconception is that latent optimization becomes sufficient once a scalar predictor such as drag is available. The framework criticizes such scalar-only latent optimization because it reduces interpretability and weakens fine-scale control; GANO instead optimizes field-level objectives computed from predicted physical fields (Sun et al., 6 May 2026).

2. Geometry representation and field surrogate

The geometry model in GANO is an implicit signed distance auto-decoder, StableSDF. A shape is represented by

$$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$

and the corresponding surface is

$$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$

Each training geometry $\Gamma_i$ has its own learnable latent code $\mathbf z_i\in\mathbb R^d$, while decoder weights are shared across the dataset. The training objective is

$$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$

with latent perturbation

$\Gamma$0

This is not an occupancy or voxel representation; it is an implicit SDF parameterization with a denoising-style latent perturbation mechanism (Sun et al., 6 May 2026).

The field predictor is GI-Transolver, a geometry-injected version of Transolver. Its key modification is the injection of geometry latent code into slice-token space. In the base Transolver, slice assignments and slice tokens are formed by

$\Gamma$1

$\Gamma$2

$\Gamma$3

followed by self-attention in token space and broadcast back to point features. GI-Transolver augments this with a gated residual geometry injection:

$\Gamma$4

$\Gamma$5

The predicted outputs depend on the benchmark: complex scattered field in Helmholtz, $\Gamma$6 for airfoils, and surface pressure for vehicles (Sun et al., 6 May 2026).

This division of labor is central. StableSDF defines a differentiable latent-to-geometry map with high geometric fidelity, while GI-Transolver provides the differentiable route from geometry code to physical field. A plausible implication is that GANO should be read less as a universal optimizer in parameter space than as a geometry-conditioned optimization system in which representation and surrogate are co-designed to preserve a reliable gradient pathway (Sun et al., 6 May 2026).

3. Latent-space optimization and geometric control

At inference time, GANO freezes StableSDF and GI-Transolver and updates only the latent code $\Gamma$7. Given the current latent $\Gamma$8, the surrogate predicts the relevant field on a boundary-consistent sample set, the objective $\Gamma$9 is evaluated from predicted fields, and gradients flow by

$\Omega(\Gamma)$0

The latent update is conceptually

$\Omega(\Gamma)$1

with $\Omega(\Gamma)$2, and in implementation an Adam step is used (Sun et al., 6 May 2026).

The framework emphasizes field-derived objectives rather than direct scalar heads. In the airfoil case, the force on the control volume boundary is

$\Omega(\Gamma)$3

from which lift and drag are computed as

$\Omega(\Gamma)$4

and the coefficients are

$\Omega(\Gamma)$5

The optimization objective is

$\Omega(\Gamma)$6

with $\Omega(\Gamma)$7 and $\Omega(\Gamma)$8. For vehicles, the drag objective is pressure-based with latent regularization:

$\Omega(\Gamma)$9

with $\mathbf{u}(\cdot;\Gamma)$0 (Sun et al., 6 May 2026).

Two geometry-control mechanisms make the latent loop practical. The first is part-wise protection by null-space projection. Given constraint points $\mathbf{u}(\cdot;\Gamma)$1, the constraint Jacobian is

$\mathbf{u}(\cdot;\Gamma)$2

and the safe direction is

$\mathbf{u}(\cdot;\Gamma)$3

This enforces $\mathbf{u}(\cdot;\Gamma)$4, so first-order changes at protected parts vanish (Sun et al., 6 May 2026).

The second is remeshing-free projection. After a latent update, previous query points drift off the new surface. Rather than reconstructing a mesh, GANO projects points back to the zero level set using

$\mathbf{u}(\cdot;\Gamma)$5

For airfoils, the implementation uses a modified version that preserves reference SDF values around the initial mesh. This makes geometry processing local, differentiable in practice, and free of repeated remeshing (Sun et al., 6 May 2026).

4. Theoretical properties

The paper’s main theoretical claim is that denoising-style latent perturbation in StableSDF induces an implicit Jacobian regularization. Fixing $\mathbf{u}(\cdot;\Gamma)$6 and target $\mathbf{u}(\cdot;\Gamma)$7, define

$\mathbf{u}(\cdot;\Gamma)$8

The perturbed objective is

$\mathbf{u}(\cdot;\Gamma)$9

The appendix shows that if $\mathcal{J}$0 is twice continuously differentiable with bounded Hessian, then near accurate reconstruction

$\mathcal{J}$1

The stated interpretation is that the noisy $\mathcal{J}$2 loss behaves like an added penalty on $\mathcal{J}$3, thereby reducing decoder sensitivity to latent perturbations (Sun et al., 6 May 2026).

A second result links reduced latent sensitivity to controlled surface movement. Under the assumptions $\mathcal{J}$4 on the surface and $\mathcal{J}$5 near the surface,

$\mathcal{J}$6

with $\mathcal{J}$7 the Hausdorff distance. This gives a local bound on geometric deformation induced by latent updates (Sun et al., 6 May 2026).

The third theoretical component concerns detached geometry updates. Writing

$\mathcal{J}$8

the full gradient decomposes as

$\mathcal{J}$9

GANO omits the transport term and differentiates only through the explicit latent-to-field path. The appendix bounds the mismatch by the norm of the omitted transport contribution and gives a tighter bound when latent sensitivity is small. This suggests that the denoising mechanism does more than stabilize decoding: it also reduces bias introduced by detaching the geometry projection path (Sun et al., 6 May 2026).

The null-space projector is given a direct optimization interpretation. The appendix proves that $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$0 is the orthogonal projection of $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$1 onto $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$2, equivalently the solution of

$$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$3

The remeshing-free projection step is likewise justified as a Gauss-Newton step for minimizing the squared level-set residual, with one-step quadratic residual contraction under regularity assumptions (Sun et al., 6 May 2026).

5. Empirical performance and computational profile

GANO is evaluated on three benchmarks spanning 2D Helmholtz, 2D airfoil aerodynamics, and 3D vehicles. On Helmholtz, the dataset contains 1000 random shapes, each simulated under 10 incident angles on a $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$4 grid. GANO achieves Rel. L1 $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$5 and Rel. L2 $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$6, better than AeroGTO at $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$7 and Transolver at $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$8. In inversion from sparse boundary observations, it reconstructs finer geometric detail than the reported differentiable baselines, and the appendix states that even with as few as 10 sensors the recovered global silhouette remains correct (Sun et al., 6 May 2026).

On AirFoil_9k, GANO achieves Rel. L1 $$s = s_{\theta}(\mathbf{x},\mathbf{z}), \qquad \mathbf{x}\in\mathbb{R}^{3},\ \mathbf{z}\in\mathbb{R}^{d},\ s\in\mathbb{R},$$9 and Rel. L2 $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$0, versus AeroGTO $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$1 and Transolver $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$2. The optimization study reports, after COMSOL validation, the following lift and drag coefficients: the initial airfoil has $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$3, $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$4, and $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$5; DeepONet reaches $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$6, $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$7, and $$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$8 ($$\Gamma(\mathbf z)=\{\mathbf x \mid s_\theta(\mathbf x,\mathbf z)=0\}.$$9); U-Net reaches $\Gamma_i$0, $\Gamma_i$1, and $\Gamma_i$2 ($\Gamma_i$3); and GANO reaches $\Gamma_i$4, $\Gamma_i$5, and $\Gamma_i$6 ($\Gamma_i$7). The reported interpretation is that GANO yields more aerodynamically reasonable optimized airfoils while preserving a smooth leading edge (Sun et al., 6 May 2026).

On DrivAerNet++, GANO achieves Rel. L1 $\Gamma_i$8 and Rel. L2 $\Gamma_i$9, compared with AeroGTO $\mathbf z_i\in\mathbb R^d$0, Transolver $\mathbf z_i\in\mathbb R^d$1, and Transolver++ $\mathbf z_i\in\mathbb R^d$2. High-fidelity OpenFOAM verification reports drag coefficient reduction from $\mathbf z_i\in\mathbb R^d$3 to $\mathbf z_i\in\mathbb R^d$4 for an estateback ($\mathbf z_i\in\mathbb R^d$5) and from $\mathbf z_i\in\mathbb R^d$6 to $\mathbf z_i\in\mathbb R^d$7 for a fastback ($\mathbf z_i\in\mathbb R^d$8). The paper notes that, unlike PhysGen, GANO preserves practical structures such as side mirrors through null-space projection (Sun et al., 6 May 2026).

The computational profile is also reported. On a single A100, full-round times are 3.93 s for Helmholtz inversion, 35.69 s for airfoil optimization, and 7.39 s for vehicle optimization. For remeshing-free projection, projecting $\mathbf z_i\in\mathbb R^d$9 points onto the updated surface takes about 30 ms in 5 iterations. On 100 vehicles with $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$0 points each, the success rate after projection reaches $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$1 under absolute SDF threshold $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$2, $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$3 under $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$4, $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$5 under $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$6, and $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$7 under $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$8. Noise-scale ablations on StableSDF report nearly unchanged reconstruction quality across $$\min_{\theta,\{\mathbf{z}_i\}} \sum_{i=1}^{N}\; \mathbb{E}_{(\mathbf{x},s)\sim\mu_i} \Big[ \big|\, s_{\theta}(\mathbf{x},\tilde{\mathbf{z}}_i) - s \,\big| \Big] \;+\; \lambda \|\mathbf{z}_i\|_2^2,$$9, with F1 around $\Gamma$00–$\Gamma$01 (Sun et al., 6 May 2026).

6. Relation to the broader geometry-aware optimization literature

The term “geometry-aware” spans several distinct research programs. GANO (Sun et al., 6 May 2026) is geometry-aware because geometry itself is represented, differentiated, and optimized. A related but different notion appears in PI-GANO, which augments a DCON-style physics-informed neural operator with a geometry encoder so that the learned operator

$\Gamma$02

generalizes across both PDE parameters and domain geometries without paired FEM data (Zhong et al., 2024). That work is a forward operator model; GANO adds an explicit latent-space optimization or inversion loop on top of a geometry-conditioned surrogate (Sun et al., 6 May 2026).

A second meaning concerns parameter-space geometry. FISMO formulates optimizer updates as a trust-region problem under a Kronecker-factored Fisher metric and applies orthogonalized momentum in whitened coordinates,

$\Gamma$03

thereby making update geometry curvature-aware rather than geometry-as-object aware (Xu et al., 29 Jan 2026). FedSPDnet addresses yet another layer: manifold-aware aggregation for Stiefel-constrained SPDnet parameters in federated learning, using ProjAvg and RLAvg to preserve orthogonality during server-side aggregation (Pautrel et al., 24 Apr 2026). In both cases, geometry refers to the parameter manifold or local metric, not to shape optimization (Xu et al., 29 Jan 2026, Pautrel et al., 24 Apr 2026).

A third meaning is symmetry-aware or non-Euclidean optimization. GAEA treats architecture parameters in weight-sharing NAS as living on a product of simplices and replaces Euclidean logit optimization by exponentiated gradient updates derived from mirror descent (Li et al., 2020). More recently, symmetry-compatible optimizer design proposes that each matrix-valued parameter block should receive an update equivariant under its own symmetry group, leading to spectral, one-sided spectral, row-norm, hybrid, and centered row-norm updates for ordinary matrices, embeddings, SwiGLU blocks, and MoE routers (Lau et al., 18 May 2026). These works are directly relevant to the phrase “geometry-aware neural optimizer,” but they operate on neural parameter blocks rather than on PDE-governed geometries (Li et al., 2020, Lau et al., 18 May 2026).

This suggests that GANO occupies a specific niche within a broader family of geometry-aware methods: it is closest to differentiable shape optimization and inversion, whereas related work distributes geometry-awareness across operator learning, manifold-constrained optimization, mirror geometry, Fisher geometry, and symmetry-compatible update design (Sun et al., 6 May 2026, Zhong et al., 2024, Xu et al., 29 Jan 2026, Pautrel et al., 24 Apr 2026, Li et al., 2020, Lau et al., 18 May 2026).

7. Limitations and open directions

The stated scope of GANO is steady-state PDE settings; spatiotemporal dynamics are not addressed. Its effectiveness depends on the learned shape manifold and training distribution, and it is therefore best suited to interpolative or near-manifold optimization rather than arbitrary extrapolation. The representation is an implicit SDF auto-decoder, and topology change is not discussed as a supported capability. The theoretical guarantees are local, relying on smoothness and regular level-set assumptions, and the remeshing-free projection is likewise local (Sun et al., 6 May 2026).

The framework also retains the standard latent-optimization sensitivities of learned shape models, even though StableSDF explicitly mitigates them. Objective design remains task-specific, initialization matters, and performance depends on the reliability of the field surrogate. The paper positions GANO against prior latent methods that drift off the shape manifold or optimize a scalar objective head, but it does not claim unrestricted global optimization guarantees (Sun et al., 6 May 2026).

Several broader directions are implied by neighboring work. PI-GANO suggests a route toward geometry-aware operator learning under variable PDE parameters and geometries without solution supervision (Zhong et al., 2024). FISMO indicates that local metric structure can be integrated into optimizer design through Fisher-informed whitening and trust-region reasoning (Xu et al., 29 Jan 2026). Symmetry-compatible blockwise optimization suggests that parameter-space symmetry and shape-space geometry might eventually be handled within one unified framework (Lau et al., 18 May 2026). A plausible implication is that future GANO-style systems could combine latent geometry optimization, geometry-conditioned field surrogates, and symmetry- or metric-aware parameter optimization in a single multi-level geometry-aware training stack.

In its present form, however, GANO is most accurately characterized as an end-to-end differentiable framework for PDE-driven shape optimization and inversion that couples an implicit SDF auto-decoder, a geometry-injected field surrogate, denoising-induced latent regularization, null-space-based part protection, and remeshing-free reprojection. Within that scope, it provides a concrete answer to a longstanding obstacle in differentiable design: how to keep geometry inside the optimization loop without collapsing either geometric fidelity or gradient reliability (Sun et al., 6 May 2026).