Internalizing Geometric Law: Learning from Solver Residuals for Precision-Critical Generation

• URL: http://arxiv.org/abs/2606.09278v1
• Date: Mon, 08 Jun 2026 09:44:31 GMT
• Title: Internalizing Geometric Law: Learning from Solver Residuals for Precision-Critical Generation
• Abstract summary: We study open-ended geometric synthesis from natural language.<n>We release PyGeoX, a programmable geometric DSL that compiles declarative constraints into a differentiable loss.<n>We propose Saturating Additive Rewards (SAR), which decompose the reward into bounded per-constraint terms.
• Score: 2.459492253868525
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Large Language Models frequently hallucinate in precision-critical domains such as technical diagramming and mechanical design, where outputs must satisfy strict geometric constraints. We study open-ended geometric synthesis from natural language: translating free-form descriptions into precise constructions whose entities must simultaneously satisfy dozens of interacting constraints. To make this tractable, we release PyGeoX, a programmable geometric DSL that compiles declarative constraints into a differentiable loss, and PyGeoX-Bench, a stratified suite of 300 problems with per-constraint verifiable rewards. Using PyGeoX as a verifier, we identify a failure mode we call Outlier Gradient Masking: under global-norm rewards (any scheme that aggregates residuals through a single norm, for example, $\exp(-\mathrm{MSE})$), a single outlier constraint can nullify the learning signal across all others. To address this, we propose Saturating Additive Rewards (SAR), which decompose the reward into bounded per-constraint terms, preserving partial progress and ensuring consistent gradients even under severe violations. Against MSE-based rewards, the natural baseline for geometry solvers, SAR improves the hard-tier solving rate by $2.3\times$, and the resulting 8B model is competitive with much larger frontier systems on this benchmark. We release the engine, benchmark, and data at https://github.com/Huawei-AI4Math/PyGeoX.

Related papers

• VeriGeo: Controllable Geometry Question Generation with Numerical and Analytical Verification [12.221274109590176]
  We introduce VeriGeo, a controllable geometry generation framework grounded in executable reasoning traces.<n>A three-stage pipeline checks numerical consistency, analytical realizability, and global consistency.<n>Supervised fine-tuning on 8.7k examples generated by VeriGeo achieves the best reported GeoQA performance.
  arXiv  Detail & Related papers  (2026-06-12T06:59:44Z)
• Mesh Graph Neural Network Framework for Accelerating Finite Element Simulation for Arbitrary Geometries [0.126928839393823]
  This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries.<n>Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks.<n>The model achieves $R2 geq 0.97$ on an unseen geometry and unseen load, compared to $R2 approx 0.01$--$0.86$ for conventional models.
  arXiv  Detail & Related papers  (2026-06-06T18:17:08Z)
• Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interaction [59.47026381165585]
  Vision-language models solve geometry problems with rising accuracy, yet their intermediate states remain latent and unverifiable.<n>We present Draw2Think, a framework that recasts geometric reasoning from latent spatial inference into agentic interaction with the GeoGebra constraint engine.
  arXiv  Detail & Related papers  (2026-05-20T05:46:14Z)
• Learning Hierarchical and Geometry-Aware Graph Representations for Text-to-CAD [22.163726324892256]
  We propose a hierarchical and geometry-aware graph as an intermediate representation.<n>Instead of mapping text directly to code, our framework first predicts structure and constraints, then conditions action sequencing and code generation.<n>Our method consistently outperforms existing approaches in both geometric fidelity and accurate satisfaction of geometric constraints.
  arXiv  Detail & Related papers  (2026-04-11T07:49:16Z)
• GIFT: Bootstrapping Image-to-CAD Program Synthesis via Geometric Feedback [16.42151644490948]
  We introduce GIFT, a framework that leverages geometric feedback to turn test-time compute into a bootstrapped set of high-quality training samples.<n>It improves mean IoU by 12% over a strong supervised baseline and remains competitive with more complex multimodal systems.
  arXiv  Detail & Related papers  (2026-03-28T23:49:20Z)
• Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later [4.346172386676835]
  We introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses.<n>We test our method on: steady-state heat over rectangular domains; Reynold-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies.
  arXiv  Detail & Related papers  (2025-09-24T09:38:11Z)
• Disentangled Representation Learning with the Gromov-Monge Gap [65.73194652234848]
  Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning.<n>We introduce a novel approach to disentangled representation learning based on quadratic optimal transport.<n>We demonstrate the effectiveness of our approach for quantifying disentanglement across four standard benchmarks.
  arXiv  Detail & Related papers  (2024-07-10T16:51:32Z)
• Universal Online Learning with Gradient Variations: A Multi-layer Online Ensemble Approach [57.92727189589498]
  We propose an online convex optimization approach with two different levels of adaptivity. We obtain $mathcalO(log V_T)$, $mathcalO(d log V_T)$ and $hatmathcalO(sqrtV_T)$ regret bounds for strongly convex, exp-concave and convex loss functions.
  arXiv  Detail & Related papers  (2023-07-17T09:55:35Z)
• $\texttt{FedBC}$: Calibrating Global and Local Models via Federated Learning Beyond Consensus [66.62731854746856]
  In federated learning (FL), the objective of collaboratively learning a global model through aggregation of model updates across devices tends to oppose the goal of personalization via local information. In this work, we calibrate this tradeoff in a quantitative manner through a multi-criterion-based optimization. We demonstrate that $texttFedBC$ balances the global and local model test accuracy metrics across a suite datasets.
  arXiv  Detail & Related papers  (2022-06-22T02:42:04Z)
• Self-supervised Geometric Perception [96.89966337518854]
  Self-supervised geometric perception is a framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels. We show that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.
  arXiv  Detail & Related papers  (2021-03-04T15:34:43Z)
• An Integer Linear Programming Framework for Mining Constraints from Data [81.60135973848125]
  We present a general framework for mining constraints from data. In particular, we consider the inference in structured output prediction as an integer linear programming (ILP) problem. We show that our approach can learn to solve 9x9 Sudoku puzzles and minimal spanning tree problems from examples without providing the underlying rules.
  arXiv  Detail & Related papers  (2020-06-18T20:09:53Z)

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.