Variational Boundary Tensors

• Variational boundary tensors are tensorial objects that encode boundary conditions in variational principles, ensuring well-posedness in physical and mathematical models.
• They are used across fields—such as gravitational theories, tensor networks, and PDE optimization—to systematically eliminate unwanted boundary terms and extract physical data.
• Their application improves numerical methods, facilitates conservation law derivations, and enhances the convergence of learning-based PDE solvers.

Variational boundary tensors are tensorial objects or expressions arising from the systematic treatment of boundary conditions, boundary contributions, or interface structures within variational formulations in physics, mathematics, and numerical analysis. They encapsulate the canonical or physically meaningful response to variations “at the boundary” in systems described by functionals, PDEs, or tensor networks. Their rigorous construction is essential for well-posed action principles and accurate modeling of boundary effects, whether in gravitational theory, the calculus of variations, PDE-constrained optimization, or quantum many-body methods.

1. Variational Principles and Boundary Terms

A variational principle is well-posed if, for a fixed set of boundary data, its stationary points correspond to the physically admissible solutions. In systems governed by functionals involving higher derivatives or constrained degrees of freedom, naively varying the action produces unwanted surface (boundary) terms. The elimination or control of these terms defines the structure of variational boundary tensors.

For gravitational theories, particularly those with second or higher derivatives in the metric, the surface variation leads not only to variations of the field but inevitably to normal derivatives at the boundary. Well-posedness requires the systematic addition of boundary terms—exemplified by the Gibbons–Hawking–York (GHY) boundary term in general relativity, or its generalizations in $F(R)$ or scalar-tensor theories—to allow the variational principle to be satisfied for natural (Dirichlet) boundary data (0809.4033). More generally, in the calculus of variations on jet bundles, the structure of the first variation splits naturally into an interior Euler–Lagrange term and a boundary Euler operator, the latter being the precise variational boundary tensor for that system (Moreno et al., 2013).

2. Boundary Tensors in Geometric Field Theories

In geometric field theories, particularly metric theories of gravity, variational boundary tensors formalize the intrinsic and extrinsic data required for a well-posed variational principle. For the Einstein–Hilbert action, the GHY term

$$S_\text{GHY} = 2 \int_{\partial \mathcal{V}} d^{n-1}x \sqrt{|h|} K$$

removes unwanted normal derivative variations by completing the boundary term produced by varying the Ricci scalar (0809.4033). For higher-derivative theories, such as $F(R)$ gravity, the correct boundary term is dictated by mapping the system to its scalar–tensor equivalent and is of the form

$$S_\text{boundary} = 2\int_{\partial\mathcal{V}} d^{n-1}x \sqrt{|h|} F'(R) K.$$

These terms are the variational boundary tensors for the respective field theories; their necessity emerges from the requirement that the variational problem does not over-constrain the degrees of freedom at the boundary, and their form is essential for recovering conserved quantities (such as ADM energy) and black hole thermodynamics (0809.4033).

A broader geometric context involves relative jet bundles: the relative Euler operator $E^\mathrm{rel}$ maps Lagrangians to pairs $(E_{\mathcal{Y}}(L), E^\partial(L))$, with the boundary part $E^\partial(L)$ representing the boundary tensor that must vanish for a solution to be extremal when free boundary conditions are allowed (Moreno et al., 2013). These objects are functorially natural and accommodate arbitrary manifold topology, order of Lagrangian, and system dimensionality.

3. Variational Boundary Tensors in Constraint and Hamiltonian Systems

In constrained systems, notably those amenable to Dirac's theory and phase space analysis, the selection of boundary data must align with the true physical degrees of freedom rather than the full configuration space. Imposing naive Dirichlet conditions overconstrains the system because certain combinations of variables are non-dynamical. The canonical approach identifies the physical (orthogonal) variables by suitable canonical transformations (Shanmugadhasan procedure), and the boundary variational tensor reduces to the terms involving these variables alone—extraneous variations are nullified by the constraints (Izumi et al., 2023).

The boundary contribution to the action is, generically,

$$p_i\, \delta q^i \approx p_{\perp,a}\, \delta q^a_\perp + \delta W,$$

where $q^a_\perp$ denote physical variables, and $W$ is an optional surface counterterm. The variational boundary tensor, in this context, precisely encodes which combinations of canonical variables should be fixed at the boundary for a consistent variational procedure (Izumi et al., 2023).

4. Boundary Tensors in Tensor Networks and Quantum Many-Body Systems

In quantum many-body and statistical systems, variational boundary tensors manifest as boundary matrix product states (MPS), corner transfer matrix (CTM) structures, or more general boundary environments used in real-space renormalization or tensor network coarse-graining.

An important example is the use of variational MPS to approximate the dominant eigenvector (environment) of the transfer matrix in infinite tensor networks (Song et al., 14 Aug 2025). The canonical form of the MPS—obtained with the VUMPS algorithm—serves as the globally optimized variational boundary tensor. These boundary tensors are then used to construct renormalization projectors (isometries) for coarse-graining, significantly improving the accuracy of tensor renormalization group (TRG) methods, especially near criticality, while maintaining favorable computational scaling (Song et al., 14 Aug 2025).

In another context, the variational boundary tensors formalize boundary and corner observables. By introducing auxiliary tensors and density matrices that preserve edge and corner degrees of freedom, modified CTMRG algorithms enable accurate measurement of local magnetizations and their derivatives at the boundary, thereby determining boundary critical exponents with high precision (Krcmar et al., 20 Jun 2025).

Boundary tensors also play a role in optimizing representations at the closure (“boundary”) of the tensor network variety, enabling efficient approximation of states otherwise inaccessible with finite bond dimension TNS by explicit augmentation with “weight states” and polynomial expansions (Christandl et al., 2020).

5. Variational Boundary Methods in Machine Learning and Optimization

Variational boundary tensors are central in enforcing boundary conditions in learning-based variational solvers for PDEs. Standard deep Ritz methods enforce boundary conditions via penalty terms, leading to nonconvexities and potential violations of the boundary constraints. In contrast, structured ansatzes constructed as

$y(x) \approx B(x) + p(x)\,N_\text{net}(x,\theta),$

with $B(x)$ interpolating boundary data and $p(x)$ vanishing at the boundary, inherently encode the required variational boundary tensor structure—removing the need for penalization and achieving both improved accuracy and faster convergence in variational optimization (Florencio et al., 18 May 2025).

6. Role in Quantifying and Controlling Boundary Contributions

Practical implementations of variational boundary tensors ensure the correct extraction of boundary conditions in field theories (e.g., extraction of shear force and bending moment conditions in plate and beam models (Schöberl et al., 2018)), boundary-energy-momentum complexes in asymptotically flat spacetimes (providing Bondi mass and angular-momentum loss equations through Ward identities (Hartong et al., 8 May 2025)), and rigorous boundary alignment in domain decomposition for mesh generation and materials science via higher-order $Q$-tensor mechanisms (Golovaty et al., 2019).

Boundary tensors are also central to developing entropy-based, dimension-free deviation inequalities for random matrices and tensors, where the variational principle links entropy with MGFs, providing sharp operator-norm bounds and statistical guarantees in high-dimensional applications (Zhivotovskiy, 2021).

7. Extensions Across Mathematical Physics

The concept of variational boundary tensors extends to double-foliation formalisms for gravity, allowing systematic splitting of metric variations into orthogonal and tangential components relative to null or more general boundaries (Aghapour et al., 2018), as well as to new approaches for general constraints, relative cohomology in geometric calculus of variations (Moreno et al., 2013), and the definition of canonical conjugate pairs and energy-momentum content in quantum gravity path integrals (Feng et al., 2021).

A key insight is that for well-posedness, boundary tensors must precisely match the physical (gauge-invariant or unconstrained) data. Generic boundary anomalies (e.g., in local Carroll boosts for asymptotically flat spacetimes) reflect deep features of underlying symmetries or anomalies in the quantum theory (Hartong et al., 8 May 2025).

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Table: Representative Occurrences of Variational Boundary Tensors

Setting	Role of Boundary Tensors	Cited Papers
Higher-derivative and modified gravity	Ensure well-posedness, encode correct Hamiltonian & entropy	(0809.4033)
Geometric calculus of variations (jet bundles)	Intrinsic formulation of natural boundary conditions	(Moreno et al., 2013, Schöberl et al., 2018)
Constrained Hamiltonian systems	Isolate physical boundary data by canonical transformation	(Izumi et al., 2023)
Tensor network coarse-graining	Encode global environment, optimize projectors	(Song et al., 14 Aug 2025, Krcmar et al., 20 Jun 2025)
Learning-based PDE solvers	Structured ansatz guaranteeing exact boundary satisfaction	(Florencio et al., 18 May 2025)
Asymptotically flat spacetimes	Define boundary energy-momentum-news complex, anomalies	(Hartong et al., 8 May 2025)

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Variational boundary tensors thus provide a unifying principle and toolkit for rigorously managing boundary contributions across a wide array of physical, mathematical, and computational domains. Their precise structure is dictated by the variational principle, the symmetry and constraint content of the system, and, in modern computational applications, by the specific methods used for optimization, coarse-graining, or numerical solution.