arXiv subjects

Dongzhe Zheng

Publications and source records attributed to Dongzhe Zheng.

At least 19 recordsLinked to original sources

arXiv2026

Mirror Symmetry of Spencer-Hodge Decompositions in Constrained Geometric Systems: Selected Collisions and Weyl Splittings

We formulate mirror symmetry for compatible principal-bundle systems with a genuinely graded polynomial Spencer complex. Exact parity identifies its Hodge spectra, while gauge implementation is governed by eigenbundle transport. For parameterized families, a spectral coefficient extension realizes the eigenline incidence as its projectivized harmonic zero modes. A relative symplectic suspension separates scalar curvature from flag energy. On a fixed spatial bundle, mixed-Chern orthogonality constrains parameter-sphere charges; a nonzero two-band charge precludes contactness over a closed base. This incidence viewpoint organizes three kinds of spectral degeneration. Continuous diagonalization across an arbitrary closed nowhere-dense collision set is controlled by flag extension, permutation monodromy, and eigenline Chern classes. Smooth flat gaps can conceal flag behavior from every jet; polynomial gaps yield finite-jet semialgebraic blow-ups. Local Dirichlet minimality then selects a rigid cone. We classify the minimizing radial cones into every $\mathbb{C}P^N$: the nonconstant links are isometric projective-line hedgehogs. This gives projective singular-measure stability and, together with a Grassmannian degree bound, power convergence at all three-dimensional minimizing singularities in these targets. For complexified four-band pencils the same incidence is read through geometric-rank minors, giving complex isolation and a degree-twenty translation cover in a crystalline model. For every spin, jointly polynomial Hermitian unfoldings attain both universal Weyl-point count bounds; constant spin-$3/2$ translations have two sixteen-point chambers with different gap distributions. Throughout, eigenflags and geometric rank retain information absent from the scalar characteristic polynomial.

math.GM↗

arXiv2026

Geometric Duality Between Constraints and Gauge Fields: Mirror Realization and Reduction Geometry on Principal Bundles

A connection and a nonzero parallel adjoint field determine an invariant hyperplane constraint on a principal bundle. Its sign mirror preserves the hyperplane and reverses its coorientation; global gauge realization is controlled by a twisted stabilizer reduction. For regular fields we identify the normalizing gauge extension as a pushout of the torus-normalizer extension, giving exact lift orders and simultaneous-splitting criteria. In singular rank-two block families, reductions on a fixed trivial bundle form an affine second-Chern lattice whose Weyl stabilizers and finite-order lift spectra detect topology invisible to paired curvature. The reduction framework also determines the structure group and second cohomology of the matched-flag diagonalization space of Friedman and Park, and gives a first- and second-Chern criterion for normal matrices with fixed separated spectrum on four-complexes; every integral solution of their three-eigenline equation on $S^2\times S^2$ is realized. For moving reductions, the projected circle curvature differs from the ambient paired curvature by a covariant-derivative term. Full fatness on a closed four-manifold forces a nontrivial sign-mirror obstruction for every circle reduction; hyperbolic self-dual-form bundles also provide circle reductions in the $y$-fat setting of Florit and Ziller. Contact transgression, bundle automorphism twists, and the natural first-jet Spencer operator complete the geometric picture.

math.GM↗

arXiv2026

Dynamical Geometry of Principal Bundle Constrained Systems: Compatible Pairs, Contact Degeneration, and Adapted Metrics

We study invariant codimension-one constraints on principal bundles through compatible pairs: a constraint distribution and a nonzero coadjoint field parallel for a principal connection. Pairing the field with the connection gives an invariant one-form whose Levi form separates a horizontal curvature contribution from a vertical coadjoint-orbit contribution. This decomposition yields criteria for integrability, contactness, and characteristic reduction; holonomy and stabilizer reductions describe global existence. For evolving compatible pairs, we identify the mixed-curvature obstruction to compatibility and prove that connection transport preserves the Levi geometry. For initially contact data over a closed base of dimension $2n$, we establish matching bounds for the quadratic $H^{n+1}$ cost of contact degeneration: making the paired curvature vanish on a Darboux ball of radius $r$ in time $T$ costs an amount comparable to $[T\log(R_*/r)]^{-1}$. The constructed paths remain contact before $T$, preserve the curvature class, and force the $L^\infty$ norm of every transporting velocity gradient to grow at least as $1/[2(T-t)]$ on the collapsing region in Darboux coordinates. On a closed three-manifold, a contact form preserved by a locally free circle action admits an invariant adapted metric with any prescribed positive curl eigenvalue and unit circle generator. We parametrize all such metrics and prove that their space is contractible. Along the circle-bundle degeneration paths, every continuous tensor limit of normalized adapted metrics is degenerate above the collapsing region.

math.GM↗

arXiv2026

Constructing Two Metrics for Hodge Theory of Constrained Bundles: Harmonic Stability and Magnetic Coercivity

We study two metric constructions for Hodge theory on bundles, together with the response of harmonic representatives and magnetic energies to degeneration. Our first result is an $L^1$ coercivity estimate for the covariant derivative on functions of bounded variation: the coercivity constant is positive exactly when the connection has no nonzero parallel section. Extension across internal interfaces then proves the uniform magnetic frustration conjecture of Chakradhar, Gittins, Habib, and Peyerimhoff, and yields positivity of both magnetic Cheeger constants on connected manifolds. For weighted de Rham complexes, we relate the defect in cohomological energy to changes in harmonic representatives. Bracket-generating harmonic vector fields give a quantitative recovery estimate with no lower bound on the varying positive weight. On the Heisenberg nilmanifold, the resulting $H^{-1/2}$ estimate has optimal Sobolev exponent. For circle bundles over surfaces, two connection metrics have identical cohomological Hodge pairings. A curvature upper bound gives a uniform positive Hodge spectral gap even as the adapted metrics lose rank. We identify the scalar spectral limit in every circle character; zero-density regions contribute magnetic Dirichlet-to-Neumann energies that retain their flat holonomy.

math.GM↗

arXiv2026

Learning Reduced-Order Dynamics with Singularity via Latent-Augmented Neural Ordinary Differential Equations

This paper addresses the issue of self-intersecting trajectories (in phase space) in industrial reduced-order modeling and proposes the Latent-Augmented Neural Ordinary Differential Equations (LA-NODEs) framework. From the perspective of artificial intelligence, the proposed method augments conventional neural ordinary differential equations to enhance model expressiveness, enabling the representation of conflicting vector fields that may arise in reduced-order systems, thereby improving learning accuracy. Through theoretical analysis, the underlying mechanism of the framework is established, and a condition for determining the minimum required augmentation dimension is derived. From the perspective of engineering applications, the effectiveness of the proposed method is validated on the reduced-order system of two representative industrial models, namely an interior permanent magnet synchronous motor (IPMSM) drive and a distributed energy system (DES). Experimental results demonstrate that the proposed method can recover system features that are difficult to capture using conventional approaches and achieve superior performance in terms of prediction accuracy and modeling fidelity, thereby providing an effective approach for high-precision data-driven modeling of complex industrial systems.

cs.LG↗

arXiv2026

Topology-Preserving Neural Operator Learning via Hodge Decomposition

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

cs.LG↗

arXiv2026

Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc

Physical fields on meshes require a separation between topology and geometry: conservation laws are topological and should be exact, while geometry, material response, and anisotropic coupling must be learned from data. Existing neural surrogates often mix these roles inside unconstrained message passing. We introduce Riemannian Hodge Message Passing (RHMP), which turns this separation into an architectural principle. RHMP fixes the cellular coboundaries ($d_k$) determined by oriented incidence and learns symmetric positive-definite cochain metrics ($H_k$) for geometry-dependent propagation. Treating $H_k$ as the learned metric motivates cochain-frame equivariance: physical propagation should be invariant to orthogonal changes of the hidden cochain feature basis. RHMP implements this principle with metric-weighted Hodge blocks ($d_k^\top H_{k+1}d_k$), yielding exact cochain-complex identities ($d_{k+1}d_k=0$), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance. Across seven physical benchmarks spanning fluids, electromagnetism, gauge fields, and variable-mesh CFD, RHMP achieves the best overall performance, with the largest gains when topology, learned geometry, and field structure interact.

cs.LG↗

arXiv2026

Neural Field Thermal Tomography: A Differentiable Physics Framework for Non-Destructive Evaluation

Inverse problems for stiff parabolic partial differential equations (PDEs), such as the inverse heat conduction problem (IHCP), are severely ill-posed: the forward map rapidly damps high-frequency interior structure before it reaches the boundary. Soft-constrained physics-informed neural networks (PINNs), which embed the PDE as a residual penalty, suffer from gradient pathology in this regime and tend to fit boundary measurements while leaving the interior field essentially untouched. We propose Neural Field Thermal Tomography (NeFTY), a hard-constrained neural field framework for label-free three-dimensional inverse heat conduction. NeFTY represents the unknown diffusivity as a continuous coordinate-based neural network, and at every optimization step passes the candidate field through a differentiable implicit-Euler heat solver with harmonic-mean interface flux, so that the governing PDE holds exactly on the discretization rather than as a soft penalty. Adjoint gradients propagate the surface reconstruction error back to the network weights at solver-level memory cost, making test-time inversion tractable on a single GPU. Across synthetic 3D benchmarks, NeFTY substantially outperforms soft-constrained PINN variants and a voxel-grid baseline on label-free volumetric recovery, and it transfers to real thermography data, surpassing classical signal-processing baselines in both defect segmentation and depth estimation. Additional details at https://cab-lab-princeton.github.io/nefty/

cs.LG↗

arXiv2026

HodgeCover: Higher-Order Topological Coverage Drives Compression of Sparse Mixture-of-Experts

Sparse Mixture-of-Experts (MoE) layers route tokens through a handful of experts, and learning-free compression of these layers reduces inference cost without retraining. A subtle obstruction blocks every existing compressor in this family: three experts can each be pairwise compatible yet form an irreducible cycle when merged together, so any score that ranks experts on pairwise signals is structurally blind to which triples are jointly mergeable. We show the obstruction is a precise mathematical object, the harmonic kernel of the simplicial Laplacian on a 2-complex whose vertices are experts, whose edges carry KL merge barriers, and whose faces carry triplet barriers; Hodge-decomposing the edge-barrier signal isolates the kernel exactly. We turn the diagnostic into a selection objective: HodgeCover greedily covers the harmonic-critical edges and triplet-critical triangles, and a hybrid variant of HodgeCover pairs it with off-the-shelf weight pruning on survivors. On three open-weight Sparse MoE backbones under aggressive expert reduction, HodgeCover matches state-of-the-art learning-free baselines on the expert-reduction axis, leads on the aggressive-compression frontier of the hybrid axis, and uniquely balances retained mass across all four Hodge components. These results show that exposing the harmonic kernel of a learned MoE structure changes which compressor wins at the regime that matters most.

cs.LG↗

arXiv2026

A Posteriori Second-Order Guarantees for Bolza Problems via Collocation

Direct collocation for Bolza optimal control yields discrete Karush-Kuhn-Tucker (KKT) points, while practical solvers expose only discrete quantities such as primal-dual iterates, reduced Hessians, and Jacobians. This creates a gap between continuous second-order optimality theory and what can be certified from solver output. We develop an a posteriori certification framework that bridges this gap. Starting from a discrete KKT solution, we reconstruct piecewise polynomial state, control, and costate trajectories, evaluate residuals of the dynamics, boundary, and stationarity conditions, and derive a computable lower bound for the continuous second variation. The bound is expressed as the discrete reduced curvature minus explicit residual-dependent correction terms. A positive bound yields a sufficient certificate for continuous second-order sufficiency and provides quantitative information relevant to local growth and trust-region sizing. The constants entering the certification inequality are conservatively estimable from reconstructed discrete data. The resulting test is operationally verifiable from collocation outputs and naturally supports adaptive mesh refinement through residual decomposition. We also outline an extension to path inequalities with isolated transversal switches.

math.OC↗

arXiv2026

Memory-Augmented Potential Field Theory: A Framework for Adaptive Control in Non-Convex Domains

Stochastic optimal control methods often struggle in complex non-convex landscapes, frequently becoming trapped in local optima due to their inability to learn from historical trajectory data. This paper introduces Memory-Augmented Potential Field Theory, a unified mathematical framework that integrates historical experience into stochastic optimal control. Our approach dynamically constructs memory-based potential fields that identify and encode key topological features of the state space, enabling controllers to automatically learn from past experiences and adapt their optimization strategy. We provide a theoretical analysis showing that memory-augmented potential fields possess non-convex escape properties, asymptotic convergence characteristics, and computational efficiency. We implement this theoretical framework in a Memory-Augmented Model Predictive Path Integral (MPPI) controller that demonstrates significantly improved performance in challenging non-convex environments. The framework represents a generalizable approach to experience-based learning within control systems (especially robotic dynamics), enhancing their ability to navigate complex state spaces without requiring specialized domain knowledge or extensive offline training.

cs.RO↗

arXiv2026

Picard groups of completed period images and the Deng-Robles problem

A basic problem in the geometry of degenerating period maps is to determine whether their completed images admit an intrinsic algebraic description. For polarized variations of Hodge structure over smooth quasi-projective surfaces, Deng and Robles formulated such a problem in terms of the Kato-Nakayama-Usui completion of the period image and a conjectural Proj description involving the augmented Hodge line bundle and the boundary divisor on a smooth compactification of the base. We show that the essential obstruction to this description is divisor-theoretic: it may be expressed as a Picard-generation statement on the completed mixed period image. We prove this statement when the pure period image is one-dimensional, and consequently obtain the Deng-Robles Proj description in this case.

math.AG↗

arXiv2025

Multiscale Cross-Modal Mapping of Molecular, Pathologic, and Radiologic Phenotypes in Lipid-Deficient Clear Cell Renal CellCarcinoma

Clear cell renal cell carcinoma (ccRCC) exhibits extensive intratumoral heterogeneity on multiple biological scales, contributing to variable clinical outcomes and limiting the effectiveness of conventional TNM staging, which highlights the urgent need for multiscale integrative analytic frameworks. The lipid-deficient de-clear cell differentiated (DCCD) ccRCC subtype, defined by multi-omics analyses, is associated with adverse outcomes even in early-stage disease. Here, we establish a hierarchical cross-scale framework for the preoperative identification of DCCD-ccRCC. At the highest layer, cross-modal mapping transferred molecular signatures to histological and CT phenotypes, establishing a molecular-to-pathology-to-radiology supervisory bridge. Within this framework, each modality-specific model is designed to mirror the inherent hierarchical structure of tumor biology. PathoDCCD captured multi-scale microscopic features, from cellular morphology and tissue architecture to meso-regional organization. RadioDCCD integrated complementary macroscopic information by combining whole-tumor and its habitat-subregions radiomics with a 2D maximal-section heterogeneity metric. These nested models enabled integrated molecular subtype prediction and clinical risk stratification. Across five cohorts totaling 1,659 patients, PathoDCCD reliably recapitulated molecular subtypes, while RadioDCCD provided reliable preoperative prediction. The consistent predictions identified patients with the poorest clinical outcomes. This cross-scale paradigm unifies molecular biology, computational pathology, and quantitative radiology into a biologically grounded strategy for preoperative noninvasive molecular phenotyping of ccRCC.

q-bio.QM↗

arXiv2025

Mirror Duality in a Spencer-Type Complex: Analytic and Riemann-Roch Perspectives

We introduce and analyze a Spencer-type elliptic complex on the space of differential forms valued in symmetric powers of an adjoint bundle, $Ω^\bullet(X)\otimes \mathrm{Sym}^\bullet(G)$. The complex is governed by a total differential $D_{λ,ψ}$ depending on a section $ψ\inΓ(G)$ and a real parameter $λ$. The central result of this paper is an algebraic realization of mirror-type duality and parameter robustness at the \emph{chain-level}. We demonstrate that sign flips ($λ\mapsto -λ$ or $ψ\mapsto -ψ$) and rescaling ($λ\mapsto αλ$) of the deformation parameters correspond to simple conjugations of the differential $D_{λ,ψ}$ by elementary zero-order automorphisms. This provides a unified, conceptual foundation for the invariance of topological invariants that is often established via case-by-case analytic methods. Analytically, this framework implies the invariance of harmonic space dimensions under the mirror map $ψ\mapsto -ψ$. Algebraically, the Grothendieck--Riemann--Roch index formula for the complex's hypercohomology is shown to be manifestly independent of $(λ, ψ)$, determined solely by the characteristic classes of a universal virtual bundle. The theory is fully compatible with equivariant localization and is verified with concrete applications on Calabi--Yau backgrounds, including K3 surfaces and elliptic curves. This framework thus offers a rigorous, chain-level explanation for the parameter robustness intrinsic to Witten-type deformations and localization phenomena, grounding them in a fundamental algebraic conjugation principle.

math.GM↗

arXiv2025

Exceptional Symmetry as a Source of Algebraic Cycles: Non-Constructive Methods for the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds

Classical variational Hodge structure theory characterizes the algebraicity of Hodge classes by studying the transversality of period mappings under geometric deformations. However, when algebraic varieties lack appropriate deformation families, this method faces applicability limitations. This paper develops a non-constructive method based on exceptional Lie group constraints to handle such cases. Our main technical contribution is establishing a dimension control mechanism for Spencer cohomology theory under Lie group constraints. Specifically, we prove that when a compact Kähler manifold $X$ is equipped with $E_7$ group constraints, the corresponding Spencer kernel $\mathcal{K}_λ^{1,1}$ has complex dimension simultaneously constrained by two bounds: representation theory gives the lower bound $\dim_{\mathbb{C}}\mathcal{K}_λ^{1,1} \geq 56$, while the degenerate Spencer-de Rham mapping gives the upper bound $\dim_{\mathbb{C}}\mathcal{K}_λ^{1,1} \leq h^{1,1}(X)$. For 5-dimensional Calabi-Yau manifolds satisfying $h^{1,1}(X) = 56$, this dimension constraint becomes the equality $\dim_{\mathbb{C}}\mathcal{K}_λ^{1,1} = 56 = h^{1,1}(X)$. Combined with our established Spencer-calibration equivalence principle, this dimension matching result is sufficient to verify the $(1,1)$-type Hodge conjecture. Our proof completely avoids explicit algebraic cycle construction, instead achieving the goal through abstract dimensional arguments. This method demonstrates the application potential of Lie group representation theory in algebraic geometry, providing new theoretical tools for handling geometric objects where traditional deformation methods are not applicable.

math.GM↗

arXiv2025

ArtGS:3D Gaussian Splatting for Interactive Visual-Physical Modeling and Manipulation of Articulated Objects

Articulated object manipulation remains a critical challenge in robotics due to the complex kinematic constraints and the limited physical reasoning of existing methods. In this work, we introduce ArtGS, a novel framework that extends 3D Gaussian Splatting (3DGS) by integrating visual-physical modeling for articulated object understanding and interaction. ArtGS begins with multi-view RGB-D reconstruction, followed by reasoning with a vision-language model (VLM) to extract semantic and structural information, particularly the articulated bones. Through dynamic, differentiable 3DGS-based rendering, ArtGS optimizes the parameters of the articulated bones, ensuring physically consistent motion constraints and enhancing the manipulation policy. By leveraging dynamic Gaussian splatting, cross-embodiment adaptability, and closed-loop optimization, ArtGS establishes a new framework for efficient, scalable, and generalizable articulated object modeling and manipulation. Experiments conducted in both simulation and real-world environments demonstrate that ArtGS significantly outperforms previous methods in joint estimation accuracy and manipulation success rates across a variety of articulated objects. Additional images and videos are available on the project website: https://sites.google.com/view/artgs/home

cs.RO↗

arXiv2025

The Rigidity of Constraint: A Spencer-Hodge Theoretic Approach to the Hodge Conjecture

This paper proposes a new theoretical perspective for studying the Hodge conjecture through an analytical framework based on constraint geometry. Our theory begins with a key observation: in compatible pair Spencer theory, a "differential degeneration" mechanism simplifies Spencer differential operators to classical exterior differential under specific algebraic conditions, bridging constraint geometry and de Rham cohomology. This bridge alone is insufficient to filter rare Hodge classes with algebraicity. We introduce the core concept -- "Spencer hyper-constraint conditions," a constraint system from Lie algebraic internal symmetry integrating: differential degeneration, Cartan subalgebra constraints, and mirror stability. This constraint principle filters geometric objects with excellent properties from degenerate classes, constructively defined as "Spencer-Hodge classes." To reveal their geometric significance, we "complex geometrize" the framework, integrating with Variation of Hodge Structures theory to construct Spencer-VHS theory. We establish connections between Spencer hyper-constraint conditions and flatness of corresponding sections under Spencer-Gauss-Manin connections. With the "Spencer-calibration equivalence principle" and "dimension matching strong hypothesis," flatness directly corresponds to algebraicity, providing sufficient conditions for verifying the Hodge conjecture. This forms "Spencer-Hodge verification criteria," transforming the proof problem into investigating whether three structured conditions hold: geometric realization of Spencer theory, satisfaction of algebraic-dimensional control, and establishment of the Spencer-calibration equivalence principle. This framework provides new perspectives for understanding this fundamental problem.

math.GM↗

arXiv2025

Extension Research of Principal Bundle Constraint System Theory on Ricci-flat Kähler Manifolds

This paper extends the geometric mechanics theory of constraint systems on principal bundles from the flat connection case to the general situation with non-zero curvature. Based on the theoretical foundation of compatible pairs under strong transversality conditions and principal bundle constraint systems, we systematically study the behavior of compatible pair theory in non-flat geometry within the context of Ricci-flat Kähler manifolds. Through rigorous mathematical analysis, we prove that the fundamental framework of strong transversality conditions and compatible pairs possesses universality and does not depend on the flatness assumption of connections. Furthermore, we re-derive the dynamical connection equations incorporating curvature reaction terms and extend Spencer cohomology theory to a spectral sequence structure capable of precisely encoding curvature information. The research reveals the exact action mechanism of curvature $Ω$: it plays a crucial role through the integrability condition $\text{ad}_Ω^* λ= 0$ and higher-order differentials in Spencer cohomology, while maintaining the fundamental structure of the theoretical framework unchanged. This extension not only deepens the geometric coupling theory between constraint systems and gauge fields but also provides new mathematical tools for modern gauge field theory, string theory geometry, and related topological analysis, demonstrating the profound connections between constraint geometry and classical differential geometry.

math.GM↗