arXiv subjects

Dongzhe Zheng

Publications and source records attributed to Dongzhe Zheng.

27 records · Page 2Linked to original sources

arXiv2025

Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry

The discovery of mirror symmetry in compatible pair Spencer complex theory brings new theoretical tools to the study of constrained geometry. Inspired by classical Spencer theory and modern Hodge theory, this paper establishes Spencer-Riemann-Roch theory in the context of constrained geometry, systematically studying the mirror symmetry of Spencer-Hodge decompositions and their manifestations in algebraic geometry. We utilize Serre's GAGA principle to algebraic geometrize Spencer complexes, establish coherent sheaf formulations, and reveal the topological essence of mirror symmetry through characteristic class theory. Main results include: Riemann-Roch type Euler characteristic computation formulas for Spencer complexes, equivalence theorems for mirror symmetry of Hodge decompositions at the characteristic class level, and verification of these theories in concrete geometric constructions. Research shows that algebraic geometric methods can not only reproduce deep results from differential geometry, but also reveal the intrinsic structure of mirror symmetry in constrained geometry through characteristic class analysis, opening new directions for applications of Spencer theory in constrained geometry.

math.GM↗

arXiv2025

Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry

Based on the compatible pair theory of principal bundle constraint systems, this paper discovers and establishes a complete Spencer differential degeneration theory. We prove that when symmetric tensors satisfy a $λ$-dependent kernel condition $δ_{\mathfrak{g}}^λ(s)=0$, the Spencer differential degenerates to the standard exterior differential, thus establishing a precise bridge between the complex Spencer theory and the classical de Rham theory. One of the advances in this paper is the rigorous proof that this degeneration condition remains stable under mirror transformations, revealing the profound symmetry origins of this phenomenon. Based on these rigorous mathematical results, we construct a canonical mapping from degenerate Spencer cocycles to de Rham cohomology and elucidate its geometric meaning. Finally, we demonstrate the application potential of this theory in algebraic geometry, particularly on K3 surfaces, where we preliminarily verify that this framework can systematically identify (1,1)-Hodge classes satisfying algebraicity conditions. This work provides new perspectives and technical approaches for studying algebraic invariants using tools from constraint geometry.

math.GM↗

arXiv2025

Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures

Learning unknown dynamics under environmental (or external) constraints is fundamental to many fields (e.g., modern robotics), particularly challenging when constraint information is only locally available and uncertain. Existing approaches requiring global constraints or using probabilistic filtering fail to fully exploit the geometric structure inherent in local measurements (by using, e.g., sensors) and constraints. This paper presents a geometric framework unifying measurements, constraints, and dynamics learning through a fiber bundle structure over the state space. This naturally induced geometric structure enables measurement-aware Control Barrier Functions that adapt to local sensing (or measurement) conditions. By integrating Neural ODEs, our framework learns continuous-time dynamics while preserving geometric constraints, with theoretical guarantees of learning convergence and constraint satisfaction dependent on sensing quality. The geometric framework not only enables efficient dynamics learning but also suggests promising directions for integration with reinforcement learning approaches. Extensive simulations demonstrate significant improvements in both learning efficiency and constraint satisfaction over traditional methods, especially under limited and uncertain sensing conditions.

cs.RO↗

arXiv2025

Constrained Hamiltonian Systems on Observation-Induced Fiber Bundles: Theory of Symmetry and Integrability

Classical constrained Hamiltonian theory assumes complete observability of system states, but in reality only partial state information is often available. This paper establishes a complete geometric theoretical framework for handling such incompletely observed systems. By introducing the concept of observation-induced fiber bundles, we naturally extend Dirac constraint theory to the fiber bundle setting, unifying the treatment of state constraints and observation constraints. Main results include: (1) Classification of existence conditions for observation fiber bundles based on characteristic class theory; (2) Complete characterization of Poisson structures on fiber bundles and corresponding symplectic reduction theory; (3) Geometric necessary and sufficient conditions for integrability and Lax pair construction; (4) Extension of Noether's theorem under symmetry group actions. The theoretical framework naturally encompasses a wide range of applications from classical mechanics to modern safety-critical control systems, providing a rigorous mathematical foundation for dynamical analysis under incomplete information.

math.GM↗

arXiv2024

On input-to-state stability verification of identified models obtained by Koopman operator

This paper proposes a class of basis functions for realizing the input-to-state stability verification of identified models obtained from the true system (assumed to be input-to-state stable) using the Koopman operator. The formulated input-to-state stability conditions are in the form of linear matrix inequalities. Two extensions are presented to relax the imposed restrictions on the basis functions. Several numerical examples are provided to demonstrate the efficacy of the proposed results.

eess.SY↗

arXiv2024

ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence

Neural ODEs (NODEs) are continuous-time neural networks (NNs) that can process data without the limitation of time intervals. They have advantages in learning and understanding the evolution of complex real dynamics. Many previous works have focused on NODEs in concise forms, while numerous physical systems taking straightforward forms, in fact, belong to their more complex quasi-classes, thus appealing to a class of general NODEs with high scalability and flexibility to model those systems. This, however, may result in intricate nonlinear properties. In this paper, we introduce ControlSynth Neural ODEs (CSODEs). We show that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities. In the composition of CSODEs, we introduce an extra control term for learning the potential simultaneous capture of dynamics at different scales, which could be particularly useful for partial differential equation-formulated systems. Finally, we compare several representative NNs with CSODEs on important physical dynamics under the inductive biases of CSODEs, and illustrate that CSODEs have better learning and predictive abilities in these settings.

cs.LG↗

arXiv2023

Differentiable Fluid Physics Parameter Identification Via Stirring

Fluid interactions permeate daily human activities, with properties like density and viscosity playing pivotal roles in household tasks. While density estimation is straightforward through Archimedes' principle, viscosity poses a more intricate challenge, especially given the varied behaviors of Newtonian and non-Newtonian fluids. These fluids, which differ in their stress-strain relationships, are delineated by specific constitutive models such as the Carreau, Cross, and Herschel-Bulkley models, each possessing unique viscosity parameters. This study introduces a novel differentiable fitting framework, DiffStir, tailored to identify key physics parameters via the common daily operation of stirring. By employing a robotic arm for stirring and harnessing a differentiable Material Point Method (diffMPM)-based simulator, the framework can determine fluid parameters by matching observations from both the simulator and the real world. Recognizing the distinct preferences of the aforementioned constitutive models for specific fluids, an online strategy was adopted to adaptively select the most fitting model based on real-world data. Additionally, we propose a refining neural network to bridge the sim-to-real gap and mitigate sensor noise-induced inaccuracies. Comprehensive experiments were conducted to validate the efficacy of DiffStir, showcasing its precision in parameter estimation when benchmarked against reported literature values. More experiments and videos can be found in the supplementary materials and on the website: https://sites.google.com/view/diffstir.

cs.RO↗

arXiv2023

Differentiable Cloth Parameter Identification and State Estimation in Manipulation

In the realm of robotic cloth manipulation, accurately estimating the cloth state during or post-execution is imperative. However, the inherent complexities in a cloth's dynamic behavior and its near-infinite degrees of freedom (DoF) pose significant challenges. Traditional methods have been restricted to using keypoints or boundaries as cues for cloth state, which do not holistically capture the cloth's structure, especially during intricate tasks like folding. Additionally, the critical influence of cloth physics has often been overlooked in past research. Addressing these concerns, we introduce DiffCP, a novel differentiable pipeline that leverages the Anisotropic Elasto-Plastic (A-EP) constitutive model, tailored for differentiable computation and robotic tasks. DiffCP adopts a ``real-to-sim-to-real'' methodology. By observing real-world cloth states through an RGB-D camera and projecting this data into a differentiable simulator, the system identifies physics parameters by minimizing the geometric variance between observed and target states. Extensive experiments demonstrate DiffCP's ability and stability to determine physics parameters under varying manipulations, grasping points, and speeds. Additionally, its applications extend to cloth material identification, manipulation trajectory generation, and more notably, enhancing cloth pose estimation accuracy. More experiments and videos can be found in the supplementary materials and on the website: https://sites.google.com/view/diffcp.

cs.RO↗

arXiv2023

UniFolding: Towards Sample-efficient, Scalable, and Generalizable Robotic Garment Folding

This paper explores the development of UniFolding, a sample-efficient, scalable, and generalizable robotic system for unfolding and folding various garments. UniFolding employs the proposed UFONet neural network to integrate unfolding and folding decisions into a single policy model that is adaptable to different garment types and states. The design of UniFolding is based on a garment's partial point cloud, which aids in generalization and reduces sensitivity to variations in texture and shape. The training pipeline prioritizes low-cost, sample-efficient data collection. Training data is collected via a human-centric process with offline and online stages. The offline stage involves human unfolding and folding actions via Virtual Reality, while the online stage utilizes human-in-the-loop learning to fine-tune the model in a real-world setting. The system is tested on two garment types: long-sleeve and short-sleeve shirts. Performance is evaluated on 20 shirts with significant variations in textures, shapes, and materials. More experiments and videos can be found in the supplementary materials and on the website: https://unifolding.robotflow.ai

cs.RO↗