Dongzhe Zheng

Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

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

Han Xue, Yutong Li, Wenqiang Xu et al.

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

Tao Zhong, Yixun Hu, Dongzhe Zheng et al. · princeton

We propose Neural Field Thermal Tomography (NeFTY), a differentiable physics framework for the quantitative 3D reconstruction of material properties from transient surface temperature measurements. While traditional thermography relies on pixel-wise 1D approximations that neglect lateral diffusion, and soft-constrained Physics-Informed Neural Networks (PINNs) often fail in transient diffusion scenarios due to gradient stiffness, NeFTY parameterizes the 3D diffusivity field as a continuous neural field optimized through a rigorous numerical solver. By leveraging a differentiable physics solver, our approach enforces thermodynamic laws as hard constraints while maintaining the memory efficiency required for high-resolution 3D tomography. Our discretize-then-optimize paradigm effectively mitigates the spectral bias and ill-posedness inherent in inverse heat conduction, enabling the recovery of subsurface defects at arbitrary scales. Experimental validation on synthetic data demonstrates that NeFTY significantly improves the accuracy of subsurface defect localization over baselines. Additional details at https://cab-lab-princeton.github.io/nefty/

Tao Zhong, Dongzhe Zheng, Christine Allen-Blanchette

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.

Dongzhe Zheng, Wenjie Mei

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.