A nonlinear extension of parametric model embedding for dimensionality reduction in parametric shape design

• URL: http://arxiv.org/abs/2605.11759v1
• Date: Tue, 12 May 2026 08:32:39 GMT
• Title: A nonlinear extension of parametric model embedding for dimensionality reduction in parametric shape design
• Abstract summary: Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information.<n>This paper introduces a nonlinear extension of PME, denoted NLPME.<n>The proposed framework preserves the defining principle of PME while replacing the linear reduced subspace with a nonlinear latent representation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Dimensionality reduction is essential in simulation-based shape design, where high-dimensional parameterizations hinder optimization, surrogate modeling, and systematic design-space exploration. Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information while preserving an explicit backmapping to the original design parameters. However, PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability. This paper introduces a nonlinear extension of PME, denoted NLPME. The proposed framework preserves the defining principle of PME -- geometry-driven latent variables and parameter-mediated reconstruction -- while replacing the linear reduced subspace with a nonlinear latent representation. Geometry is not reconstructed directly from the latent variables; instead, the latent representation is decoded into admissible design parameters, and the corresponding geometry is recovered through a forward parametric map. The method is assessed on a bio-inspired autonomous underwater glider with a 32-dimensional parametric shape description and a CAD-based geometry-generation process. NLPME reaches a 5\% reconstruction-error threshold with \(N=5\) latent variables, compared with \(N=8\) for linear PME, and a 1\% threshold with \(N=9\), compared with \(N=15\) for PME. Comparison with a deep autoencoder shows that most of the nonlinear compression gain can be retained while preserving an explicit backmapping to the original design variables. The results establish NLPME as a compact, admissible, and engineering-compatible nonlinear reduced representation for parametric shape design spaces.

Related papers

• Autoregressive B-Rep Shape Generation with Parametric Surfaces [83.03418080213359]
  ParaCAD is an autoregressive framework for point-cloud-conditioned B-Rep generation.<n>Our model first generates parametric surfaces with constrained UV domains, and then constructs a valid B-Rep by globally intersecting these surfaces.<n>Extensive experiments demonstrate that ParaCAD produces accurate B-Reps with faithful point-cloud alignment.
  arXiv  Detail & Related papers  (2026-07-19T06:14:50Z)
• Pointer-CAD v2: Plan-Then-Construct CAD Generation with Dimension-Aware Parametric Precision [30.320445641359466]
  Computer-aided design (CAD) plays a fundamental role in modern manufacturing by providing the high precision required for industrial production.<n>Recent large language model based approaches formulate CAD generation as a sequence prediction problem and have achieved promising results.<n>Small numerical deviations that are negligible at the shape-level may still violate industrial tolerance requirements.<n>We present Pointer-CAD v2, which directly predicts continuous values, bypassing the need for quantized numerical parameters and thereby eliminating quantization errors.
  arXiv  Detail & Related papers  (2026-06-28T09:40:20Z)
• A Geometry-Adaptive Deep Variational Framework for Phase Discovery in the Landau-Brazovskii Model [4.702925112226925]
  We propose a Geometry-Adaptive Deep Variational Framework (GeoDVF) for pattern-forming systems.<n>By explicitly treating the domain size as trainable variables within the variational formulation, GeoDVF naturally eliminates artificial stress during training.<n>We show that GeoDVF provides a robust and geometry-consistent variational solver capable of identifying both stable and metastable states without prior knowledge.
  arXiv  Detail & Related papers  (2026-03-05T13:32:23Z)
• An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry [0.6138671548064355]
  Being infinite dimensional, non-parametric information geometry has long faced an "intractability barrier"<n>This paper introduces a novel framework to resolve the intractability with an Orthogonal Decomposition of the Tangent Space.<n>By defining the Information Capture Ratio, we provide a rigorous method for estimating intrinsic dimensionality in high-dimensional data.
  arXiv  Detail & Related papers  (2025-12-25T00:18:41Z)
• Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation [79.27003481818413]
  We introduce FlatVI, a training framework that regularises the latent manifold of discrete-likelihood variational autoencoders towards Euclidean geometry.<n>By encouraging straight lines in the latent space to approximate geodesics on the decoded single-cell manifold, FlatVI enhances compatibility with downstream approaches.
  arXiv  Detail & Related papers  (2025-07-15T23:08:14Z)
• Generalized Linear Mode Connectivity for Transformers [87.32299363530996]
  A striking phenomenon is linear mode connectivity (LMC), where independently trained models can be connected by low- or zero-loss paths.<n>Prior work has predominantly focused on neuron re-ordering through permutations, but such approaches are limited in scope.<n>We introduce a unified framework that captures four symmetry classes: permutations, semi-permutations, transformations, and general invertible maps.<n>This generalization enables, for the first time, the discovery of low- and zero-barrier linear paths between independently trained Vision Transformers and GPT-2 models.
  arXiv  Detail & Related papers  (2025-06-28T01:46:36Z)
• Decoupled Geometric Parameterization and its Application in Deep Homography Estimation [52.96857897366727]
  Planar homography, with eight degrees of freedom (DOFs), is fundamental in numerous computer vision tasks.<n>This paper presents a novel geometric parameterization of homographies, leveraging the similarity- kernel-similarity decomposition for projective transformations.<n>Our proposed parameterization allows for direct homography estimation through matrix multiplication, eliminating the need for solving a linear system, and achieves performance comparable to the four-corner positional offsets in deep homography estimation.
  arXiv  Detail & Related papers  (2025-05-22T12:33:29Z)
• Shrinking: Reconstruction of Parameterized Surfaces from Signed Distance Fields [2.1638817206926855]
  We propose a novel method for reconstructing explicit parameterized surfaces from Signed Distance Fields (SDFs) Our approach iteratively contracts a parameterized initial sphere to conform to the target SDF shape, preserving differentiability and surface parameterization throughout. This enables downstream applications such as texture mapping, geometry processing, animation, and finite element analysis.
  arXiv  Detail & Related papers  (2024-10-04T03:39:15Z)
• Shape-informed surrogate models based on signed distance function domain encoding [8.052704959617207]
  We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs) Our approach is based on the combination of two neural networks (NNs)
  arXiv  Detail & Related papers  (2024-09-19T01:47:04Z)
• Maintaining Structural Integrity in Parameter Spaces for Parameter Efficient Fine-tuning [78.39310274926535]
  Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence.<n>To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner.<n>This paper introduces a generalized parameter-efficient fine-tuning framework, designed for various dimensional parameter space.
  arXiv  Detail & Related papers  (2024-05-23T16:04:42Z)
• Automatic Parameterization for Aerodynamic Shape Optimization via Deep Geometric Learning [60.69217130006758]
  We propose two deep learning models that fully automate shape parameterization for aerodynamic shape optimization. Both models are optimized to parameterize via deep geometric learning to embed human prior knowledge into learned geometric patterns. We perform shape optimization experiments on 2D airfoils and discuss the applicable scenarios for the two models.
  arXiv  Detail & Related papers  (2023-05-03T13:45:40Z)
• Non-linear Independent Dual System (NIDS) for Discretization-independent Surrogate Modeling over Complex Geometries [0.0]
  Non-linear independent dual system (NIDS) is a deep learning surrogate model for discretization-independent, continuous representation of PDE solutions. NIDS can be used for prediction over domains with complex, variable geometries and mesh topologies. Test cases include a vehicle problem with complex geometry and data scarcity, enabled by a training method.
  arXiv  Detail & Related papers  (2021-09-14T23:38:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.