VDAT: Variational Discrete Action Theory

• VDAT is a discrete variational framework that unifies various formulations—ranging from mechanics to quantum many-body theory—via a central discrete action functional.
• It employs techniques like discrete Lagrangians, constrained minimization, and multi-time formulations to derive update maps, integrable lattice equations, and optimal control laws.
• Its broad applicability enables structure-preserving simulations in classical dynamics, integrable systems, and efficient quantum and control algorithms.

Variational Discrete Action Theory (VDAT) denotes, in the cited literature, a set of discrete variational frameworks that place a discrete action functional at the center of analysis and derive dynamics, control laws, integrable lattice equations, or ground-state approximations from stationarity, constrained minimization, or related closure conditions. In one major line of work, VDAT is the higher-order, constrained, structure-preserving framework obtained by discretizing Hamilton’s principle directly; in another, it is the pluri-Lagrangian multi-time theory of surface-dependent discrete actions; in another, it is an integer-time quantum many-body formalism built from the sequential product density matrix (SPD) ansatz and a discrete action in compound space; and in another, it is a variational-inference formulation of control on discrete-event decision processes (Colombo et al., 2013, Boll et al., 2013, Cheng et al., 2020, Dong et al., 2019).

1. Terminological scope and shared variational structure

A common source of confusion is that the same acronym is used for non-equivalent constructions. The cited works collectively show that VDAT is not restricted to one mathematical object or application domain. This suggests that the unifying feature is methodological rather than disciplinary: the primitive quantity is discrete action, and the derived objects are update maps, corner equations, free-energy objectives, or integer-time Green’s functions, depending on context (Colombo et al., 2013, Lobb et al., 2013, Cheng et al., 2020, Dong et al., 2019).

Usage in the literature	Primitive discrete object	Characteristic result
Geometric mechanics	$S_d=\sum L_d$	symplectic variational integrator
Integrable lattice systems	action on discrete surfaces	closure $dL=0$ and surface independence
Discrete-event control	variational free-energy / ELBO	EP-style policy improvement
Quantum many-body theory	SPD with integer-time discrete action	SCDA exact in $d=\infty$

In the geometric-mechanics literature, VDAT is explicitly described as “the viewpoint that discrete dynamical schemes should be derived by applying Hamilton’s principle directly at the discrete level,” with a discrete Lagrangian $L_d$ approximating the action integral of a continuous Lagrangian $L$. In the multiform literature, the analogous primitive object is a discrete $d$-form whose action is evaluated on arbitrary $d$-surfaces in a higher-dimensional multi-time. In the quantum many-body literature, the primitive object is the SPD together with an integer-time discrete action and corresponding Green’s-function formalism. In the discrete-event control literature, control is recast as variational inference on trajectory distributions, with rewards entering as log-potentials (Colombo et al., 2013, Boll et al., 2013, Cheng et al., 2020, Dong et al., 2019).

2. Classical discrete mechanics, constraints, and structure preservation

In the mechanical formulation, a first-order discrete Lagrangian is $L_d:Q\times Q\to\mathbb{R}$, and for a uniform time-step $h$ the discrete action of a path $q=(q_0,\dots,q_N)$ is

$dL=0$0

Stationarity with fixed endpoints yields the discrete Euler–Lagrange equations

$dL=0$1

Under the standard regularity assumption that $dL=0$2 is invertible, the update $dL=0$3 is well posed. The associated left and right discrete Legendre transforms,

$dL=0$4

$dL=0$5

pull back the canonical symplectic form on $dL=0$6 to the same discrete two-form $dL=0$7, and the discrete flow preserves $dL=0$8 exactly. If a Lie group $dL=0$9 acts on $d=\infty$0 and $d=\infty$1 is $d=\infty$2-invariant, the discrete momentum map $d=\infty$3 is conserved along the discrete flow (Colombo et al., 2013).

The higher-order extension replaces $d=\infty$4 by $d=\infty$5. A higher-order discrete Lagrangian is $d=\infty$6, the discrete action is

$d=\infty$7

and stationarity with fixed boundary blocks $d=\infty$8 and $d=\infty$9 yields the higher-order discrete Euler–Lagrange equations

$L_d$0

For higher-order discrete constraints $L_d$1, one introduces multipliers $L_d$2 and the augmented action

$L_d$3

The resulting constrained update is implicit, but under the nonsingularity condition of Proposition 3.2 it defines a unique local map $L_d$4. The discrete Poincaré–Cartan one-forms $L_d$5 satisfy $L_d$6, so $L_d$7; after restriction to the constraint manifold, $L_d$8 is symplectic and $L_d$9. If both $L$0 and $L$1 are $L$2-invariant, the higher-order discrete momentum map is preserved (Colombo et al., 2013).

Time dependence can be incorporated by treating time as a discrete variable:

$L$3

Variation with respect to $L$4 yields a discrete energy balance. If $L$5 is autonomous, $L$6 and the discrete energy $L$7 is exactly preserved, yielding a symplectic and energy-momentum preserving method. The same line of development extends to stochastic Hamiltonian systems: stochastic discrete Hamiltonian variational integrators are derived from a stochastic action functional based on a type-II stochastic generating function, they are symplectic almost surely, preserve integrals of motion related to Lie group symmetries, include stochastic symplectic Runge–Kutta methods as a special case, and achieve global mean-square order $L$8 for scalar noise and for commutative multidimensional noise when only $L$9-type quadrature is used (Colombo et al., 2013, Holm et al., 2016).

A parallel canonical treatment emphasizes that discrete variational systems have both pre- and post-constraint surfaces. For a nearest-neighbor action $d$0, the pre- and post-momenta are

$d$1

with momentum matching $d$2 on solutions. Singular Legendre maps yield pre-constraints $d$3 and post-constraints $d$4. A notable discrete peculiarity is that first-class constraints need not generate symmetries unless they coincide as both pre and post; on evolving phase spaces, the number of constraints at a fixed step depends on the initial and final step of evolution. This formulation is explicitly designed to handle constant and evolving phase spaces, lattice field theory, and discrete gravity (Dittrich et al., 2013).

3. Pluri-Lagrangian systems and discrete integrability

In the integrable-systems literature, the relevant VDAT object is a Lagrangian multiform. A $d$5-dimensional pluri-Lagrangian problem on $d$6, with $d$7, asks for fields $d$8 such that for any oriented $d$9-dimensional manifold $d$0, the action

$d$1

is stationary with respect to all interior vertex variations. For $d$2, one uses a discrete Lagrangian $d$3-form on oriented elementary squares $d$4, and the central local equations are the corner equations supported on $d$5D corners. The discrete exterior derivative on an elementary cube is

$d$6

and the multi-time Euler–Lagrange equations are the eight equations $d$7. For three-point $d$8-forms of ABS type,

$d$9

the cube action reduces to an octahedral expression independent of $L_d:Q\times Q\to\mathbb{R}$0 and $L_d:Q\times Q\to\mathbb{R}$1, so the eight equations reduce to six nontrivial corner equations (Boll et al., 2013).

The closure relation

$L_d:Q\times Q\to\mathbb{R}$2

on solutions is the variational integrability criterion. It implies that the action depends only on the boundary of the surface, hence is invariant under local flips. The paper “What is integrability of discrete variational systems?” proves that for the three-point $L_d:Q\times Q\to\mathbb{R}$3-forms encoding the ABS list, the corner equations are consistent in the sense of minimal rank $L_d:Q\times Q\to\mathbb{R}$4 per cube, and that the corresponding $L_d:Q\times Q\to\mathbb{R}$5-forms are closed not only on solutions of the underlying quad-equations but also on general solutions of the corner equations. The same work also exhibits a pluri-Lagrangian system not coming from a multidimensionally consistent system of quad-equations, showing that the pluri-Lagrangian theory goes beyond the ABS setup (Boll et al., 2013).

A closely related formulation treats the action as depending simultaneously on the field and on the choice of discrete surface. For $L_d:Q\times Q\to\mathbb{R}$6-dimensional systems embedded in higher-dimensional lattices, the generalized Euler–Lagrange equations include both ordinary field variations and local surface variations, and the closure relation around an elementary cube enforces surface independence. For $L_d:Q\times Q\to\mathbb{R}$7-dimensional systems, a Lagrangian $L_d:Q\times Q\to\mathbb{R}$8-form on oriented cubes satisfies an analogous hypercube closure condition. This is the multiform version of multidimensional consistency, and the paper explicitly argues that the variational principle can be considered as the defining equations for the Lagrangians themselves (Lobb et al., 2013).

A further development concerns convex variational principles for discrete Laplace-type equations induced by integrable quad-equations. On the white graph of a bipartite quad-graph, the generalized discrete action

$L_d:Q\times Q\to\mathbb{R}$9

has Euler–Lagrange equations

$h$0

Under the reality conditions derived in the paper, $h$1 becomes real; for Q1–Q3 the sign or unit-circle conditions on the labels are necessary and sufficient for strict convexity or concavity, while for Q4 they are sufficient. Convexity then gives existence and uniqueness for Dirichlet boundary value problems. In the Q3 case, the functional is variationally equivalent to the Bobenko–Springborn circle-pattern functionals, relating discrete integrability, geometry, and convex action principles (Bobenko et al., 2011).

4. Optimal control, control-as-inference, and data-driven variational learning

A control-theoretic usage of VDAT appears in discrete variational optimal control. Here the forced discrete Lagrange–d’Alembert principle uses discrete forces

$h$2

and yields the forced discrete Euler–Lagrange equations

$h$3

The optimal control problem is then reformulated as a variational integrator of an augmented, higher-dimensional discrete Lagrangian system on spaces such as $h$4, $h$5, or nonholonomic analogues. The resulting discrete necessary conditions are again Euler–Lagrange equations, and therefore inherit the preservation properties of variational integrators on configuration manifolds, Lie groups, underactuated systems, and nonholonomic systems with symmetries (Jimenez et al., 2012).

A distinct usage appears in discrete-event control of complex systems. There, VDAT is defined as a variational framework for optimal control in systems whose interventions are discrete actions or events. The dynamical model is a Discrete Event Decision Process (DEDP) with event intensities

$h$6

trajectory distribution

$h$7

and control-as-inference target

$h$8

Approximate inference is carried out with a Bethe free-energy approximation and an EP-style forward–backward message-passing algorithm on projected singleton kernels, with overall per policy-evaluation sweep $h$9. In the reported transportation benchmarks, VDAT on SynthTown achieved TRPE $q=(q_0,\dots,q_N)$0 and EC $q=(q_0,\dots,q_N)$1, versus GPS $q=(q_0,\dots,q_N)$2 and EC $q=(q_0,\dots,q_N)$3, AC $q=(q_0,\dots,q_N)$4 and EC $q=(q_0,\dots,q_N)$5, and PG $q=(q_0,\dots,q_N)$6 and EC $q=(q_0,\dots,q_N)$7; on Berlin, VDAT achieved TRPE $q=(q_0,\dots,q_N)$8 and EC $q=(q_0,\dots,q_N)$9, while GPS, AC, and PG had negative rewards and failed to converge within a reasonable number of epochs (Dong et al., 2019).

A data-driven extension appears in “Variational Learning of Euler-Lagrange Dynamics from Data.” Although the authors do not use the term VDAT, their approach is explicitly characterized as relying on discrete Euler–Lagrange constraints derived from a discrete action, on variational integrators for prediction, and on variational backward error analysis. The discrete action is

$dL=0$00

with midpoint or trapezoidal $dL=0$01, and the learning target is an inverse modified Lagrangian $dL=0$02 satisfying the chosen discrete variational integrator. The method uses only position data, compensates discretization errors by inverse variational backward error analysis, and yields improved long-horizon energy behavior on the pendulum and Hénon–Heiles systems. In the pendulum experiment, the modified energy $dL=0$03 along LSI trajectories remained within an oscillation band of width $dL=0$04, versus $dL=0$05 for $dL=0$06 (Ober-Blöbaum et al., 2021).

5. Quantum many-body VDAT: SPD, integer time, and SCDA

In the quantum many-body literature, VDAT is a variational theory for ground-state properties of quantum Hamiltonians built from the sequential product density matrix

$dL=0$07

where $dL=0$08, $dL=0$09 are bilinear variational parameters, and $dL=0$10 are interacting projectors. The integer $dL=0$11 labels “integer time,” meaning the ordered sequence index of the SPD rather than physical time. The variational manifold is nested, so increasing $dL=0$12 monotonically improves the approximation, and $dL=0$13 recovers the exact solution (Cheng et al., 2020).

The hierarchy reproduces familiar ansätze at small $dL=0$14. The cited works state that $dL=0$15 recovers Hartree–Fock; $dL=0$16 recovers Gutzwiller, Baeriswyl, Jastrow, and unitary/variational coupled cluster; and $dL=0$17 recovers Gutzwiller–Baeriswyl and Baeriswyl–Gutzwiller. The quantum VDAT formalism then introduces an integer-time interaction representation, a compound-space discrete action, a generating function $dL=0$18, an integer-time Dyson equation,

$dL=0$19

and an integer-time Bethe–Salpeter structure for two-particle correlators. This construction generalizes path-integral and Green’s-function techniques to integer time and makes SPD evaluation tractable (Cheng et al., 2020).

Two exact limits are central. First, for impurity-like local projectors, SPD-l can be exactly evaluated by summing a finite number of integer-time diagrams. For the one-bath-site Anderson impurity model with

$dL=0$20

the variational minimum yields

$dL=0$21

which is the exact ground-state energy for that problem. Second, for the Hubbard model in $dL=0$22, the self-consistent canonical discrete action approximation (SCDA) is the integer-time analogue of DMFT and exactly evaluates SPD-d by assuming locality of the integer-time self-energy. In this sense, VDAT offers a variational hierarchy that is exact in $dL=0$23 while retaining a cost profile far below continuous-time impurity solvers (Cheng et al., 2020).

The earlier proposal “Variational Discrete Action Theory” emphasizes the practical consequence of this hierarchy: $dL=0$24 exactly reproduces the Gutzwiller approximation in $dL=0$25, whereas $dL=0$26, which exactly evaluates generalized Gutzwiller–Baeriswyl states, already provides a “truly minimal yet precise description of Mott physics with a cost similar to the GA.” The same work explicitly presents SCDA update equations for $dL=0$27, $dL=0$28, and $dL=0$29, making the analogy with DMFT structural rather than metaphorical (Cheng et al., 2020).

6. Multi-orbital algorithms, qubit parametrization, and the 1RDM functional

Subsequent many-body developments concentrate on multi-orbital Hubbard models in $dL=0$30. “Precise ground state of multi-orbital Mott systems via the variational discrete action theory” introduces a decoupled minimization algorithm for $dL=0$31–$dL=0$32. The paper states that $dL=0$33 rigorously recovers the multi-orbital Gutzwiller approximation, while $dL=0$34 “precisely captures the competition between the Hubbard $dL=0$35, Hund $dL=0$36, and crystal field $dL=0$37 in the two orbital Hubbard model over all parameter space, with a negligible computational cost.” For sufficiently large $dL=0$38 and $dL=0$39, $dL=0$40 drives a first-order transition within the Mott insulating regime, and in the large orbital-polarization limit with finite $dL=0$41, interactions remain nontrivial even for small $dL=0$42. The same work reports that $dL=0$43 can be run in seconds on a single CPU core for two orbitals (Cheng et al., 2022).

The gauge-constrained $dL=0$44 algorithm sharpens SCDA by using a gauge constraint that automatically satisfies the self-consistency condition on integer-time Green’s functions. Closed-form expressions are derived for general density-density interactions, allowing straightforward application up to the seven-orbital Hubbard model. Reported single-core timings for one SPD evaluation at $dL=0$45 range from $dL=0$46 for $dL=0$47 to $dL=0$48 at $dL=0$49. In the SU$dL=0$50 benchmarks, the $dL=0$51 critical interaction $dL=0$52 closely follows the DMFT fit, whereas $dL=0$53 gives a larger $dL=0$54 (Cheng et al., 2023).

A later reformulation maps the local Hilbert space to a $dL=0$55-qubit system with

$dL=0$56

and rewrites the $dL=0$57 VDAT trial energy in terms of the momentum density distribution, the shape of a reference Fermi surface, and a pure qubit state. For the SU$dL=0$58 Hubbard model, this qubit parametrization yields an explicit procedure for computing $dL=0$59 and gives the large-$dL=0$60 asymptotic expression

$dL=0$61

The same paper shows that the qubit parametrization applies also to $dL=0$62, where the G-type variant yields an identical expression to slave-spin mean-field theory (Cheng et al., 18 Aug 2025).

The same year, the constrained-search program was pushed to the level of one-body reduced density-matrix functional theory. Using the $dL=0$63 VDAT ansatz, the authors explicitly construct a 1RDM functional for the multi-orbital Hubbard model in the thermodynamic limit and report that “non-analytic behavior emerges in our 1RDMF at fixed integer filling, which gives rise to the Mott transition.” They further explain this by separating the constrained search into multiple stages and show how a nonzero Hund exchange drives the continuous Mott transition to become first-order. This use of VDAT is noteworthy because it translates the discrete-action variational hierarchy into an explicit orbital functional capable of representing Mott and Hund physics up to seven orbitals (Cheng et al., 18 Aug 2025).

Taken together, these developments indicate a mature quantum-many-body branch of VDAT: $dL=0$64 reproduces GA; $dL=0$65 is the practical high-accuracy regime; gauge fixing and qubit parametrization reduce SCDA to explicit local optimization problems; and the same structure can be repackaged as a 1RDM functional. A plausible implication is that, within this literature, VDAT serves both as a variational hierarchy and as a constructive route from wave-function ansätze to explicit functionals.