Dirac–Frenkel–Onsager Dynamics

• Dirac–Frenkel–Onsager Dynamics are variational evolution schemes that project time-dependent PDEs onto nonlinear manifolds while handling gauge freedom through Onsager minimization.
• They employ instantaneous residual minimization via a least-squares formulation to optimally select the tangent space representation of the evolving system.
• Extensions such as gauge momentum and inertial formulations address nullspace ambiguity and improve numerical stability in complex parameter regimes.

Dirac–Frenkel–Onsager dynamics denote a class of variational evolution schemes in which a time-dependent PDE or abstract evolution equation is projected onto a nonlinear trial manifold by the Dirac–Frenkel principle, while non-uniqueness, ill-conditioning, or dissipation is handled by an Onsager-type minimization or relaxation structure. In the 2026 formulation, the defining feature is that classical Dirac–Frenkel instantaneous residual minimization is retained at the function level, whereas gauge freedom in parameter space is resolved by a history variable injected only along Jacobian-nullspace directions; a closely related inertial formulation replaces exact gauge fixing by a second-order relaxation of the parameter velocity (Raviola et al., 30 Apr 2026, Raviola et al., 23 Jun 2026). The same variational architecture also appears in Onsager–Rayleighian continuum models of charge-regulated electrolytes and in neural-network approximations of Onsager–Machlup dynamics, which broaden the notion from a specific algorithm to a general projected non-equilibrium formalism (Zheng et al., 2024, Li et al., 2023).

1. Dirac–Frenkel projection on nonlinear manifolds

The common starting point is an evolution equation on a Hilbert space, written as

$\partial_t u(t,x)=\mathcal F\big(u(t,\cdot)\big),$

or, in abstract form,

$\partial_t u(t)=F(u(t)).$

The solution is approximated by a nonlinear parametrization

$$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$

with trial manifold

$$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$

By the chain rule,

$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$

so the time derivative is constrained to the tangent space of $\mathcal M$ (Raviola et al., 30 Apr 2026).

The classical Dirac–Frenkel principle chooses, at each time, the tangent vector that instantaneously minimizes the PDE residual in $L^2$: $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$ In parameter space this becomes the least-squares problem

$$\dot\theta(t)\in \operatorname*{arg\,min}_{\eta\in\mathbb R^p} \big\| \nabla_\theta \hat u(\theta(t),\cdot)^\top \eta - \mathcal F(\hat u(\theta(t),\cdot)) \big\|_{L^2(\Omega)}^2.$$

Equivalently, with Gram matrix and right-hand side

$$G_{ij}(\theta)=\langle \partial_{\theta_i}\hat u,\partial_{\theta_j}\hat u\rangle, \qquad g_i(\theta)=\langle \partial_{\theta_i}\hat u,\mathcal F(\hat u)\rangle,$$

the normal equations are

$\partial_t u(t)=F(u(t)).$0

This formulation is exact at the level of the chosen manifold: it does not solve the full evolution equation in ambient function space, but it does select the best tangent approximation in the metric used by the least-squares problem. The central issue addressed by Dirac–Frenkel–Onsager dynamics is that this function-space statement does not by itself determine a unique or well-conditioned parameter ODE whenever the parametrization is redundant or nearly singular.

2. Gauge freedom, nullspaces, and tangent-space collapse

The defining structural observation is that the Dirac–Frenkel condition uniquely specifies the function-level derivative $\partial_t u(t)=F(u(t)).$1 but may fail to specify a unique parameter velocity. If $\partial_t u(t)=F(u(t)).$2 is singular, all Dirac–Frenkel-compatible parameter velocities have the form

$\partial_t u(t)=F(u(t)).$3

where $\partial_t u(t)=F(u(t)).$4 is any chosen reference solution, typically the minimal-norm one. Because

$\partial_t u(t)=F(u(t)).$5

for $\partial_t u(t)=F(u(t)).$6, these nullspace directions do not change the time derivative of the represented function to first order. The 2026 DFO formulation interprets this parameter-level non-uniqueness as a gauge freedom (Raviola et al., 30 Apr 2026).

This interpretation is consequential because it separates two distinct pathologies. First, there is exact rank deficiency, where the Jacobian has a nontrivial kernel. Second, there is near rank deficiency, where the singular values of $\partial_t u(t)=F(u(t)).$7 decay strongly and the least-squares inversion becomes numerically unstable. In both cases the function-level Dirac–Frenkel condition may remain meaningful while parameter velocities become erratic, non-unique, or artificially frozen.

The wave-collision toy model in the 2026 DFO paper makes this distinction explicit. Two Gaussian pulses are parametrized by four parameters. At collision, the gradient vectors $\partial_t u(t)=F(u(t)).$8 and $\partial_t u(t)=F(u(t)).$9 become identical, the tangent space drops in dimension, and the “separation direction” $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$0 lies in the nullspace. Minimal-norm Dirac–Frenkel then assigns zero velocity in that direction, so the waves remain locked together after collision. The advection–reaction toy model exhibits a related phenomenon: when both partial derivatives vanish, the tangent space collapses to $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$1, the minimal-norm velocity becomes zero, and the dynamics freezes.

A common misconception is that such failures show a defect in the Dirac–Frenkel principle itself. The 2026 analysis argues instead that the failure is parameter-level: the function-level tangent minimization remains well defined, but the reduced coordinates leave a gauge underdetermined. Dirac–Frenkel–Onsager dynamics act precisely on that underdetermination.

3. Onsager gauge fixing and inertial generalizations

The Onsager component enters by adding a secondary variational principle in parameter space. In the gauge-momentum formulation, one introduces a history variable $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$2 with tracking energy

$$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$3

and quadratic dissipation potential

$$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$4

Onsager’s minimum-dissipation principle yields

$$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$5

so $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$6 is an exponential moving average of the reference Dirac–Frenkel velocity. Gauge fixing is then formulated as a quadratic minimization over the nullspace of $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$7, producing

$$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$8

where $$\hat u(t,x)=\hat u(\theta(t),x), \qquad \theta(t)\in\mathbb R^p,$$9 is the orthogonal projector onto $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$0 and $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$1 controls the strength of the nullspace motion (Raviola et al., 30 Apr 2026).

Because $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$2, the added term does not modify the function-level tangent derivative: $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$3 The instantaneous residual minimization is therefore preserved exactly. This is the main distinction from Tikhonov regularization or damping across all directions, which alters the least-squares solution itself and therefore biases the Dirac–Frenkel tangent vector.

A later inertial formulation adds a second-order relaxation directly to the parameter velocity. For the Tikhonov-regularized defect

$$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$4

the acceleration $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$5 is defined by minimizing

$$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$6

which yields the first-order system

$$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$7

with

$$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$8

This formulation yields well-posed parameter dynamics and a posteriori error bounds, but it does not preserve the instantaneous Dirac–Frenkel condition exactly; instead it makes the velocity relax toward the regularized Dirac–Frenkel minimizer over a timescale $$\mathcal M=\{\hat u(\theta,\cdot):\theta\in\mathbb R^p\}.$$9 in each singular direction (Raviola et al., 23 Jun 2026).

Formulation	Evolution law	Defining property
DFO gauge momentum	$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$0, $$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$1	Preserves instantaneous residual minimization exactly
DFI inertia	$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$2, $$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$3	Well-posed parameter ODE with previous-velocity memory

In discrete time, the gauge-momentum scheme uses

$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$4

where $$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$5. The inertial scheme uses a semi-implicit Euler step whose velocity update is itself a variational problem,

$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$6

so the previous velocity appears as an anchor rather than a nullspace-only gauge. Both constructions are Onsager-type because they add a quadratic dissipation or inertia functional to the projected dynamics, but they do so in fundamentally different subspaces.

4. Continuum Rayleighians and neural Onsager–Machlup manifolds

A broader use of the term arises when Onsager-type variational dynamics are projected not merely onto parameter vectors but onto coarse-grained fields or neural-network manifolds. In the charge-regulated macro-ion model, the central object is the Rayleighian

$$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$7

where $$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$8 is a Poisson–Boltzmann free energy augmented by charge regulation and $$\partial_t \hat u(\theta(t),\cdot)=\nabla_\theta \hat u(\theta(t),\cdot)^\top \eta(t),\qquad \eta(t)\equiv \dot\theta(t),$$9 is a quadratic dissipation functional in the currents of three effective mobile components: bare macro-ions, macro-ions with associated $\mathcal M$0, and free $\mathcal M$1 counter-ions. Using slow variables $\mathcal M$2, $\mathcal M$3, and $\mathcal M$4, together with a fast electrostatic potential $\mathcal M$5 satisfying $\mathcal M$6, minimization of the Rayleighian yields generalized PNP equations with explicit charge-regulation contributions. In the fixed-charge limit $\mathcal M$7, $\mathcal M$8, the formulation recovers the classical Poisson–Nernst–Planck currents and, after linearization with constant total ion density, the Debye–Falkenhagen equation

$\mathcal M$9

The detailed exposition presents this as a concrete example of a gradient-flow-like, Dirac–Frenkel–Onsager viewpoint for non-equilibrium soft matter (Zheng et al., 2024).

A complementary development is the deep Onsager–Machlup method. There the field variables, fluxes, and, when needed, stresses are represented by deep neural networks,

$L^2$0

and the loss functional is built from the Onsager–Machlup action together with penalties for boundary conditions, initial conditions, and conservation laws: $L^2$1 For diffusion, Cahn–Hilliard dynamics, and Stokes–Cahn–Hilliard flow, the Onsager–Machlup term is quadratic in the deviation of the fluxes or stresses from the corresponding Onsager drifts. The paper explicitly interprets this as a projection of Onsager–Machlup variational dynamics onto a neural-network manifold, conceptually analogous to a Dirac–Frenkel restriction of dissipative continuum dynamics (Li et al., 2023).

These continuum and neural formulations shift the emphasis from gauge freedom in a finite-dimensional parameter ODE to the more general problem of reducing a thermodynamically consistent variational dynamics onto selected slow variables or trial manifolds. This suggests that “Dirac–Frenkel–Onsager dynamics” is best understood as a family of projected non-equilibrium evolutions rather than a single algorithmic recipe.

5. Numerical behavior, applications, and limitations

The gauge-momentum formulation was designed to address singular and near-singular parameter regimes without biasing the function-level Dirac–Frenkel projection. Its analytic demonstrations are the wave-collision and advection–reaction collapse examples, where the nullspace momentum allows the dynamics to pass through exact tangent-space collapse points rather than freezing. In low-dimensional PDE experiments with neural networks, the method was evaluated on rotating detonation waves, transport through a flow field, and charged particles in an electric field. Across all three PDEs, it achieves the lowest average $L^2$2 errors and lowest final-time errors among methods that preserve Dirac–Frenkel structure, while the runtime overhead relative to DF+tSVD is described as negligible. In the 5D Fokker–Planck example, it yields smooth trajectories for mean and covariance and improves the corresponding errors over DF+tSVD, TENG, NIVP, and RSNG; a randomized SVD variant reduces runtime by approximately $L^2$3 while preserving accuracy (Raviola et al., 30 Apr 2026).

The inertial formulation addresses a different failure mode: excessive shrinkage caused by strong regularization or by compressed least-squares information. In the Allen–Cahn experiment, increasing the regularization parameter degrades Tikhonov-DF rapidly, whereas inertia remains accurate over a larger range because useful velocity information persists from past steps. In a 10D Fokker–Planck problem with importance sampling, Tikhonov-DF initially tracks the reference but later shows large errors in mean and covariance, while the inertial scheme keeps these errors small. The same qualitative advantage persists under sketching, because the transported velocity compensates for information missing from the current sketched least-squares problem (Raviola et al., 23 Jun 2026).

The two approaches also have distinct limitations. The gauge-momentum DFO method does not fix ill-conditioning in function-relevant directions directly; it addresses only gauge-induced non-uniqueness and nullspace ambiguity. It introduces the hyperparameters $L^2$4 and $L^2$5, and the paper does not provide a systematic tuning strategy. The inertial method introduces a memory trade-off: when $L^2$6, the scheme becomes less responsive to current residual information, and both the empirical results and the discrete error bounds show deterioration in that regime. A further limitation, stated explicitly for the gauge-momentum scheme, is that nullspace motion is neutral only to first order; finite time steps can therefore introduce higher-order drift in function space.

The numerical record nonetheless clarifies the operational difference between the two variants. Gauge-momentum DFO is a nullspace-restricted regularization that preserves the exact Dirac–Frenkel tangent. Inertial DFI is a previous-velocity-anchored relaxation that sacrifices exact instantaneous Dirac–Frenkel optimality in exchange for well-posedness and robustness.

6. Information geometry, reciprocity, and conceptual scope

A broader theoretical backdrop is provided by the quantum Onsager program for open quantum systems. In that setting, states $L^2$7 evolve by a quantum Markov semigroup

$L^2$8

near a steady state $L^2$9 satisfying $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$0. One considers a family of nearby initial states $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$1, defines generalized forces by

$$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$2

and represents tangent vectors through scores $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$3 determined by a density map $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$4. The quadratic expansion of a divergence $$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$5 yields a quantum Fisher information matrix

$$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$6

and its decay defines the information-loss tensor

$$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$7

The resulting rate equations are

$$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$8

with

$$\partial_t \hat u(\theta(t),\cdot) = \operatorname*{arg\,min}_{v\in \mathcal T_{\hat u(\theta)}\mathcal M} \|v-\mathcal F(\hat u(\theta(t),\cdot))\|_{L^2(\Omega)}^2.$$9

The tensor $$\dot\theta(t)\in \operatorname*{arg\,min}_{\eta\in\mathbb R^p} \big\| \nabla_\theta \hat u(\theta(t),\cdot)^\top \eta - \mathcal F(\hat u(\theta(t),\cdot)) \big\|_{L^2(\Omega)}^2.$$0 is symmetric and positive semidefinite, and under an antiunitary notion of time reversal together with a suitable detailed balance condition, the transport tensors satisfy quantum Onsager–Casimir relations of the form

$$\dot\theta(t)\in \operatorname*{arg\,min}_{\eta\in\mathbb R^p} \big\| \nabla_\theta \hat u(\theta(t),\cdot)^\top \eta - \mathcal F(\hat u(\theta(t),\cdot)) \big\|_{L^2(\Omega)}^2.$$1

depending on how the reverse preparation is defined (Tsang, 2024).

These results do not by themselves constitute a Dirac–Frenkel algorithm, but they provide the ingredients from which a quantum Dirac–Frenkel–Onsager dynamics can be built: a state manifold, a Fisher-information metric, a divergence playing the role of a Lyapunov functional, and Onsager transport tensors constrained by reciprocity. This suggests a general decomposition in which a projected Hamiltonian part supplies the Dirac–Frenkel sector, while the Fisher-metric gradient flow supplies the Onsager sector.

Within this wider perspective, Dirac–Frenkel–Onsager dynamics can be understood as a unifying variational language for reduced non-equilibrium evolution. In one branch, it yields nullspace-aware parameter dynamics for nonlinear PDE ansätze. In another, it yields Rayleighian continuum equations for charged soft matter with internal state variables. In another, it yields neural-manifold approximations to Onsager–Machlup dynamics. Across these settings, the recurring structure is the same: projection onto a chosen manifold of admissible states, supplemented by a quadratic dissipation or history principle that selects a thermodynamically or geometrically preferred evolution.