Geometry-Aware Neural Optimizer for Shape Optimization and Inversion

Geometry is central to PDE-governed systems, motivating shape optimization and inversion. Classical pipelines conduct costly forward simulation with geometry processing, requiring substantial expert effort. Neural surrogates accelerate forward analysis but do not close the loop because gradients from objectives to geometry are often unavailable. Existing differentiable methods either rely on restrictive parameterizations or unstable latent optimization driven by scalar objectives, limiting interpretability and part-wise control. To address these challenges, we propose Geometry-Aware Neural Optimizer (\textbf{\textsc{GANO}}), an end-to-end differentiable framework that unifies geometry representation, field-level prediction, and automated optimization/inversion in a single latent-space loop. \textsc{GANO} encodes shapes with an auto-decoder and stabilizes latent updates via a denoising mechanism, and a geometry-informed surrogate provides a reliable gradient pathway for geometry updates. Moreover, \textsc{GANO} supports part-wise control through null-space projection and uses remeshing-free projection to accelerate geometry processing. We further prove that denoising induces an implicit Jacobian regularization that reduces decoder sensitivity, yielding controlled deformations. Experiments on three benchmarks spanning 2D Helmholtz, 2D airfoil, and 3D vehicles show state-of-the-art accuracy and stable, controllable updates, achieving up to +55.9% lift-to-drag improvement for airfoils and ~7% drag reduction for vehicles.

Many engineering problems depend strongly on shape, such as how air flows around an aircraft wing or a car, or how waves scatter around an obstacle. Finding a better shape usually requires many rounds of expensive physical simulation, manual geometry editing, and remeshing, which makes the design process slow and difficult to automate. This work introduces GANO, a machine learning framework for automatic shape optimization and shape recovery. GANO learns a smooth digital representation of shapes, predicts the physical fields around them, and then updates the shape directly using gradients from the design objective. This allows the method to improve or reconstruct shapes without repeatedly rebuilding meshes or relying on hand-designed shape parameters. A key feature of GANO is that it can make controlled changes: for example, it can optimize a vehicle body while keeping parts such as mirrors fixed. Experiments on wave scattering, airfoil design, and 3D vehicle aerodynamics show that GANO gives accurate physical predictions and stable shape updates. In the tested cases, it improves airfoil lift-to-drag ratio by up to 55.9% and reduces vehicle drag by about 7%, suggesting a practical path toward faster, more automated engineering design.