Publications | Ting Lin

2025

arxiv

Discretizing linearized Einstein-Bianchi system by symmetric and traceless tensors

Yuyang Guo , Jun Hu, and Ting Lin

Aug 2025

Abs arXiv

The Einstein-Bianchi system uses symmetric and traceless tensors to reformulate Einstein's original field equations. However, preserving these algebraic constraints simultaneously remains a challenge for numerical methods. This paper proposes a new formulation that treats the linearized Einstein-Bianchi system (near the trivial Minkowski metric) as the Hodge wave equation associated with the conformal Hessian complex. To discretize this equation, a conforming finite element conformal Hessian complex that preserves symmetry and traceless-ness simultaneously is constructed on general three-dimensional tetrahedral grids, and its exactness is proven.
SIIMS

Analysis of a wavelet frame based two-scale model for enhanced edges

Bin Dong, Ting Lin, Peichu Xie, and Zuowei Shen

SIAM Journal on Imaging Sciences, Jul 2025

Abs arXiv PDF

Image restoration is critical across many fields, as it addresses the challenge of recovering clear images from degraded data. Two prominent approaches to this problem are wavelet-based methods and partial differential equation (PDE) models. Wavelet methods can be viewed as discrete analogs of PDE models, and through asymptotic analysis, wavelet models often converge to PDE-based approaches such as the total variation model. Wavelet methods are known for their simple implementation and multiscale time-frequency analysis, while PDE models provide a geometric interpretation of image structures, particularly edges. The relationship between these two approaches offers a comprehensive framework for image restoration. This paper designs a wavelet frame-based image restoration model, focusing on enhancing edge preservation and regularity. More importantly, we establish a connection to the $L^1$ version of the Mumford–Shah model, showing that the wavelet model converges to this variational model. This connection is significant, as it combines the geometric explanation of edges in the Mumford–Shah model with the simplicity of wavelet-based implementation. The primary contribution of this paper lies in the asymptotic analysis and proof of convergence of the two-scale wavelet model to the $L^1$ Mumford–Shah model, providing both theoretical insights and practical wavelet models for image restoration with enhanced edge detection.
SIMA

On the Universal Approximation Property of Deep Fully Convolutional Neural Networks

Qianxiao Li, Ting Lin, and Zuowei Shen

SIAM Journal on Mathematical Analysis, Jun 2025

Abs arXiv PDF

We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer counterparts can achieve universal approximation of these symmetric functions at constant channel width. Moreover, we show that the same can be achieved by nonresidual variants with at least two channels in each layer and convolutional kernel size of at least 2. In addition, we show that these requirements are necessary in the sense that networks with fewer channels or smaller kernels fail to be universal approximators.
arxiv

Mixed Finite element method for stress gradient elasticity

Ting Lin, and Shudan Tian

Jun 2025

Abs arXiv

This paper develops stable finite element pairs for the linear stress gradient elasticity model, overcoming classical elasticity's limitations in capturing size effects. We analyze mesh conditions to establish parameter-robust error estimates for the proposed pairs, achieving unconditional stability for finite elements with higher vertex continuity and conditional stability for Continuous Galerkin-Discontinuous Galerkin (CG-DG) pairs when no interior vertex has edges lying on three or fewer lines. Numerical experiments validate the theoretical results, demonstrating optimal convergence rates.
NeurIPS

A unified framework on the universal approximation of transformer-type architectures

Jingpu Cheng, Qianxiao Li, Ting Lin, and Zuowei Shen

NeurIPS, Jun 2025

Abs arXiv

We investigate the universal approximation property (UAP) of transformer-type architectures, providing a unified theoretical framework that extends prior results on residual networks to models incorporating attention mechanisms. Our work identifies token distinguishability as a fundamental requirement for UAP and introduces a general sufficient condition that applies to a broad class of architectures. Leveraging an analyticity assumption on the attention layer, we can significantly simplify the verification of this condition, providing a non-constructive approach in establishing UAP for such architectures. We demonstrate the applicability of our framework by proving UAP for transformers with various attention mechanisms, including kernel-based and sparse attention mechanisms. The corollaries of our results either generalize prior works or establish UAP for architectures not previously covered. Furthermore, our framework offers a principled foundation for designing novel transformer architectures with inherent UAP guarantees, including those with specific functional symmetries. We propose examples to illustrate these insights.
MoC

The sharpness condition for constructing a finite element from a superspline

Jun Hu, Ting Lin, Qingyu Wu, and Beihui Yuan

Mathematics of Computation, May 2025

Abs arXiv PDF

This paper addresses sharpness conditions for constructing $ C^r $ conforming finite element spaces from superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre--element spaces, which encompasses both the superspline spaces and the finite element spaces. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $ C^r $ conforming elements in $ d $ dimensions generally requires an extra $ C^{2^{s - 1}r} $ continuity on $ s $--codimensional subsimplices, and the polynomial degree is at least $ (2^d r + 1) $.
arxiv

Finite element form-valued forms: Construction

Kaibo Hu, and Ting Lin

Mar 2025

Abs arXiv

We provide a finite element discretization of $ \ell $-form-valued $ k $-forms on triangulation in $ \mathbb{R}^n $ for general $ k $, $ \ell $ and $ n $ and any polynomial degree. The construction generalizes finite element Whitney forms for the de Rham complex and their higher-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension.
SICON

Interpolation, Approximation and Controllability of Deep Neural Networks

Jingpu Cheng, Qianxiao Li, Ting Lin, and Zuowei Shen

SIAM Journal on Control and Optimization, Feb 2025

Abs arXiv PDF

We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal interpolation—the ability to match arbitrary input and target training samples—and the closely related notion of universal approximation—the ability to approximate input-target functional relationships via flow maps. Under the assumption of affine invariance of the control family, we give a characterization of universal interpolation, showing that it holds for essentially any architecture with nonlinearity. Furthermore, we elucidate the relationship between universal interpolation and universal approximation in the context of general control systems, showing that the two properties cannot be deduced from each other. At the same time, we identify conditions on the control family and the target function that ensure the equivalence of the two notions.
SIAGA

Distributional Hessian and divdiv complexes on triangulation and cohomology

Kaibo Hu, Ting Lin, and Qian Zhang

SIAM Journal on Applied Algebra and Geometry, Feb 2025

Abs arXiv PDF

We construct discrete versions of some Bernstein–Gelfand–Gelfand (BGG) complexes, i.e., the Hessian and the divdiv complexes, on triangulations in two dimensions and three dimensions. The sequences consist of finite elements with local polynomial shape functions and various types of Dirac measures on subsimplices (generalizations of currents). The construction generalizes Whitney forms (canonical conforming finite elements) for the de Rham complex and Regge calculus/finite elements for the elasticity (Riemannian deformation) complex from discrete topological and Discrete Exterior Calculus perspectives. We show that the cohomology of the resulting complexes is isomorphic to the continuous versions, and thus isomorphic to the de Rham cohomology with coefficients.

2024

FoCM

A Construction of $C^r$ Conforming Finite Element Spaces in Any Dimension

Jun Hu, Ting Lin, and Qingyu Wu

Foundations of Computational Mathematics, Dec 2024

Abs arXiv PDF

This paper proposes a construction of $C^r$ conforming finite element spaces with arbitrary r in any dimension. It is shown that if $k \ge 2^d r +1$ the space $\mathcal P_k$ of polynomials of degree $\le k$ can be taken as the shape function space of $C^r$ finite element spaces in d dimensions. This is the first work on constructing such $C^r$ conforming finite elements in any dimension in a unified way.
JMLR

Deep Neural Network Approximation of Invariant Functions through Dynamical Systems

Qianxiao Li, Ting Lin, and Zuowei Shen

Journal of Machine Learning Research, Aug 2024

Abs arXiv PDF

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions include the much studied translation-invariant ones involving image tasks, but also encompasses many permutation-invariant functions that find emerging applications in science and engineering. We prove sufficient conditions for universal approximation of these functions by a controlled dynamical system, which can be viewed as a general abstraction of deep residual networks with symmetry constraints. These results not only imply the universal approximation for a variety of commonly employed neural network architectures for symmetric function approximation, but also guide the design of architectures with approximation guarantees for applications involving new symmetry requirements.
JSC

Finite Element Grad grad Complexes and Elasticity Complexes on Cuboid Meshes

Jun Hu, Yizhou Liang, and Ting Lin

Journal of Scientific Computing, Apr 2024

Abs arXiv PDF

This paper constructs two conforming finite element grad grad and elasticity complexes on cuboid meshes. For the finite element grad grad complex, an $ H^2 $ conforming finite element space, an $ \boldsymbol{H}(\text{curl}; \mathbb{S}) $ conforming finite element space, an $ \boldsymbol{H}(\text{div}; \mathbb{T}) $ conforming finite element space and an $ \boldsymbol{L}^2 $ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector-valued $ \boldsymbol{H}^1 $ conforming space and an $ \boldsymbol{H}(\text{curl curl}^\top; \mathbb{S}) $ conforming space are constructed. Combining with an existing $ \boldsymbol{H}(\text{div}; \mathbb{S}) \cap \boldsymbol{H}(\text{div div}; \mathbb{S}) $ element and an $ \boldsymbol{H}(\text{div}; \mathbb{S}) $ element, respectively, these finite element spaces form two finite element elasticity complexes. The exactness of all the finite element complexes is proved.
C&MA

Local $C^r$ interpolation operators on simplicial triangulations in any dimension

Jun Hu, Ting Lin, and Qingyu Wu

Computers and Mathematics with Applications, Jan 2024

Abs PDF

This paper presents a construction of local $ C^r $ interpolations on simplicial grids in $ \mathbb{R}^d $ by using piecewise polynomials of degree $ k \geq 2^d r + 1 $. To this end, an interpolative decomposition of the set of corresponding multi-indices is introduced. It is shown that the piecewise polynomial interpolation is of $ C^r $ continuity provided that the corresponding function is of $ C^{2^{d - 1} r} $ continuity.

2023

arxiv

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology

Snorre Christiansen, Kaibo Hu, and Ting Lin

Dec 2023

Abs arXiv

We show that the cohomology of the Regge complex in three dimensions is isomorphic to $ \mathcal{H}_{dR}^{\bullet}(\Omega) \otimes \mathcal{RM} $, the infinitesimal-rigid-body-motion-valued de Rham cohomology. Based on an observation that the twisted de Rham complex extends the elasticity (Riemannian deformation) complex to the linearized version of coframes, connection $ 1 $-forms, curvature and Cartan's torsion, we construct a discrete version of linearized Riemann-Cartan geometry on any triangulation and determine its cohomology.
arxiv

Finite elements for symmetric and traceless tensors in three dimensions

Kaibo Hu, Ting Lin, and Bowen Shi

Nov 2023

Abs arXiv

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes. This complex includes vector fields and symmetric and traceless tensor fields, interlinked through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors and York splits in general relativity. We provide bubble complexes and investigate supersmoothness to facilitate the construction. We show the exactness of the finite element complex on contractible domains.
JEMS

Deep Learning via Dynamical Systems: An Approximation Perspective

Qianxiao Li, Ting Lin, and Zuowei Shen

Journal of the European Mathematical Society, May 2023

EASIAM Abs arXiv PDF

EASIAM best paper award, 1st prize, 2023.

We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuous-time deep residual networks, which can also be understood as approximation theories in $L^p$ using flow maps of dynamical systems. In specific cases, rates of approximation in terms of the time horizon are also established. Overall, these results reveal that composition function approximation through flow maps presents a new paradigm in approximation theory and contributes to building a useful mathematical framework to investigate deep learning.
arxiv

Local Bounded Commuting Projection Operators for Discrete Gradgrad Complexes

Jun Hu, Yizhou Liang, and Ting Lin

Apr 2023

arXiv
arxiv

Local bounded commuting projection operator for discrete de Rham complexes

Jun Hu, Yizhou Liang, and Ting Lin

Mar 2023

arXiv