Dirac–Frenkel Instantaneous Residual Minimization

• Dirac–Frenkel IRM is a variational projection principle that minimizes the instantaneous PDE residual by optimally matching the projected tangent velocity on a trial manifold.
• The approach reveals inherent gauge freedom, where non-unique parameter velocities allow for nullspace corrections without altering the function-space evolution.
• By employing techniques like truncated SVD and history-dependent momentum (DFO), the method achieves improved accuracy and stability in time-dependent simulations.

Dirac–Frenkel instantaneous residual minimization (IRM) is a variational projection principle for time-dependent approximation of evolution equations in which the velocity of an approximate solution is chosen to minimize, at each instant, the residual of the governing dynamics within the tangent space of a trial manifold. For a parametrized approximation $\hat u(\theta(t),\cdot)$ of a PDE solution, the principle selects a tangent velocity that is closest in the ambient Hilbert-space norm to the exact vector field, yielding a least-squares problem in parameter space and an orthogonality condition in function space (Raviola et al., 30 Apr 2026). In nonlinear and redundant parametrizations, however, the parameter velocity can become non-unique or numerically unstable even when the induced function-space derivative remains well defined. The 2026 Dirac–Frenkel–Onsager formulation interprets this defect as a gauge freedom and introduces nullspace-restricted momentum that preserves IRM exactly in range-space directions while regularizing only the gauge component (Raviola et al., 30 Apr 2026).

1. Variational definition and projection structure

Let $\Omega \subset \mathbb{R}^d$ be the spatial domain and $[0,T]$ the time interval. Consider an evolutionary PDE

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$

with solution in a Hilbert space $\mathcal{U}$ equipped with the $L^2$ inner product

$$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$

Given a differentiable parametrization $\hat u:\mathbb{R}^p \to \mathcal{U}$ and a time-dependent parameter vector $\theta(t)\in\mathbb{R}^p$, the chain rule gives

$$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$

Hence the approximate time derivative lies in the tangent space

$\Omega \subset \mathbb{R}^d$0

where $\Omega \subset \mathbb{R}^d$1 is the trial manifold (Raviola et al., 30 Apr 2026).

The Dirac–Frenkel principle chooses the tangent velocity that instantaneously minimizes the PDE residual in function space: $\Omega \subset \mathbb{R}^d$2 At the parameter level this becomes

$\Omega \subset \mathbb{R}^d$3

Writing $\Omega \subset \mathbb{R}^d$4 for the Jacobian operator with columns $\Omega \subset \mathbb{R}^d$5, $\Omega \subset \mathbb{R}^d$6, and $\Omega \subset \mathbb{R}^d$7, the least-squares optimality conditions are the normal equations

$\Omega \subset \mathbb{R}^d$8

with

$\Omega \subset \mathbb{R}^d$9

Equivalently, the residual is orthogonal to the tangent space: $[0,T]$0 This is the Dirac–Frenkel or TDVP orthogonality condition: the PDE dynamics are projected onto the trial manifold (Raviola et al., 30 Apr 2026).

The same projection viewpoint appears in other settings. On tensor Banach spaces, fixed Tucker-rank sets form Banach manifolds, and Dirac–Frenkel is formulated as instantaneous residual minimization on the tangent space of the fixed-rank manifold, with Hilbert-space orthogonality recovered as a special case (Falcó et al., 2016). In reduced-density-matrix geometry, the principle is expressed as orthogonal projection in the Hilbert–Schmidt metric, yielding the Bogoliubov–de-Gennes and Hartree–Fock–Bogoliubov equations as instantaneous residual minimizers on quasifree manifolds (Benedikter et al., 2017).

2. Parameter non-uniqueness as gauge freedom

A central feature of IRM for nonlinear parametrizations is the distinction between function-space uniqueness and parameter-space non-uniqueness. Even when IRM determines the function-level derivative uniquely, the parameter velocity $[0,T]$1 can be non-unique or ill-conditioned if the Jacobian has a nontrivial kernel or very small singular values (Raviola et al., 30 Apr 2026).

Let

$[0,T]$2

and let $[0,T]$3 be a reference solution of the normal equations, chosen in (Raviola et al., 30 Apr 2026) as the minimal-norm solution. Then every instantaneous residual-minimizing parameter velocity has the form

$[0,T]$4

Directions in $[0,T]$5 do not change $[0,T]$6 to first order and therefore do not violate IRM. This is interpreted as a gauge freedom: different parameter velocities represent the same tangent function-space evolution.

The numerical manifestation of this gauge structure is ill-conditioning. If $[0,T]$7 has zero singular values, tangent-space collapse occurs; if it has very small singular values, near-collapse occurs. In either case, the Gram matrix $[0,T]$8 becomes singular or nearly singular, and parameter velocities become non-unique or unstable (Raviola et al., 30 Apr 2026). In computational practice, a truncated singular value decomposition is used: $[0,T]$9 where singular values below a tolerance $$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$0 are truncated. The associated range and nullspace projectors are

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$1

This interpretation clarifies a common misconception. Non-uniqueness of $$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$2 does not imply non-uniqueness of the IRM tangent vector in function space; the ambiguity lies only in parameter-space representatives of the same tangent-space projection. A related misconception is that any regularization preserving stability also preserves IRM. Standard Tikhonov regularization,

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$3

equivalently

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$4

acts in all directions and perturbs the range-space component, thereby biasing the instantaneous residual minimizer (Raviola et al., 30 Apr 2026).

A closely related 2026 development introduces inertia rather than nullspace-only momentum. In that formulation, the Dirac–Frenkel velocity is replaced by a second-order evolution for the velocity variable, so that weakly informed directions retain past information while well-informed directions track the regularized Dirac–Frenkel solution (Raviola et al., 23 Jun 2026). This suggests that recent work has converged on a shared diagnosis: singular and near-singular parameter dynamics are best understood as an underdetermined or weakly informed tangent-space inference problem rather than as a failure of the manifold projection itself.

3. Dirac–Frenkel–Onsager dynamics

The Dirac–Frenkel–Onsager (DFO) principle augments IRM by introducing a history variable $$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$5, interpretable as momentum, and injecting it only along gauge directions. The construction is based on Onsager’s minimum-dissipation principle and is designed to preserve IRM exactly in the resolved range space (Raviola et al., 30 Apr 2026).

The mismatch energy and dissipation potential are

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$6

and $$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$7 is updated by minimizing the Rayleighian

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$8

The stationarity condition yields the linear filter

$$\partial_t u(t,x) = \mathcal{F}(u(t,\cdot)), \qquad x\in\Omega,$$9

with explicit solution

$\mathcal{U}$0

Thus $\mathcal{U}$1 is an exponential moving average of the reference IRM velocity (Raviola et al., 30 Apr 2026).

Gauge fixing is then performed by the nullspace-only optimization

$\mathcal{U}$2

which yields the closed-form system

$\mathcal{U}$3

with $\mathcal{U}$4 scaling the projected momentum.

The key structural identities are

$\mathcal{U}$5

Therefore the range-space component of the dynamics is exactly the IRM minimizer, while only the nullspace component is modified (Raviola et al., 30 Apr 2026). This is the defining difference between DFO and conventional regularization.

The paper states the preservation result explicitly: DFO dynamics strictly preserve IRM in the range space because $\mathcal{U}$6 at all times. At tangent-space collapse, the nullspace enlarges, and $\mathcal{U}$7 can supply a nonzero velocity if the history variable has accumulated nontrivial past motion. In the 1D wave equation with a two-wave parametrization, the paper proves an exact recovery statement: with $\mathcal{U}$8, $\mathcal{U}$9, and

$L^2$0

the exact crossing path

$L^2$1

satisfies the DFO dynamics for all $L^2$2, and the parametrized solution equals the exact wave solution (Raviola et al., 30 Apr 2026).

This formulation should not be conflated with standard momentum methods such as heavy-ball or Nesterov acceleration. Those methods inject momentum in all directions to improve convergence to minimizers. DFO, by contrast, injects momentum only in nullspace directions because the projected PDE dynamics must remain unchanged in range-space directions (Raviola et al., 30 Apr 2026).

4. Discretization and computational realization

In numerical implementations, the $L^2$3 inner product is approximated by quadrature or collocation. Selecting points $L^2$4, one assembles

$L^2$5

and solves the least-squares problem

$L^2$6

by truncated SVD (Raviola et al., 30 Apr 2026).

For DFO, the reference IRM velocity and nullspace projector are

$L^2$7

A semi-implicit Euler discretization combines an implicit history update with an explicit parameter update: $L^2$8

$L^2$9

with $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$0. Because $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$1 for any $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$2, $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$3 is a convex combination of $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$4 and $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$5 (Raviola et al., 30 Apr 2026).

The single-step algorithm given in (Raviola et al., 30 Apr 2026) is:

1. Assemble $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$6 and $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$7.
2. Compute tSVD: $$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$8.
3. Compute the minimal-norm IRM velocity:

$$\langle v,w\rangle := \int_\Omega v(x)\, w(x)\, dx.$$9

1. Update the history:

$\hat u:\mathbb{R}^p \to \mathcal{U}$0

2. Project onto the nullspace:

$\hat u:\mathbb{R}^p \to \mathcal{U}$1

3. Update parameters:

$\hat u:\mathbb{R}^p \to \mathcal{U}$2

The dominant cost is the tSVD of $\hat u:\mathbb{R}^p \to \mathcal{U}$3, scaling as $\hat u:\mathbb{R}^p \to \mathcal{U}$4 for $\hat u:\mathbb{R}^p \to \mathcal{U}$5. A randomized tSVD with sketch size $\hat u:\mathbb{R}^p \to \mathcal{U}$6 reduces this to $\hat u:\mathbb{R}^p \to \mathcal{U}$7, with the nullspace projector approximated from the randomized right singular subspace. The paper reports that this randomized DFO variant halves runtime in the 5D experiment (Raviola et al., 30 Apr 2026).

Practical guidelines in the paper emphasize consistent use of the same retained singular subspace both for $\hat u:\mathbb{R}^p \to \mathcal{U}$8 and for the nullspace projection, robust truncation of small singular values, and implicit history updates to avoid time-step restrictions. The authors also state several limitations. DFO does not cure ill-conditioning in function-relevant directions associated with small but nonzero singular values; those must still be managed by the least-squares solver, for example by tSVD truncation. Moreover, for nonlinear parametrizations, nullspace motion can produce higher-order changes in the represented function over finite time steps, because the construction controls first-order tangent-space dynamics by design (Raviola et al., 30 Apr 2026).

5. Singular regimes, benchmarks, and observed behavior

The paper’s illustrative examples focus on tangent-space collapse and near-collapse. In the colliding-wave example, rank loss occurs at the collision time $\hat u:\mathbb{R}^p \to \mathcal{U}$9. Minimal-norm Dirac–Frenkel cannot separate the waves after collapse, whereas DFO injects momentum along the nullspace direction $\theta(t)\in\mathbb{R}^p$0, restores motion, and recovers the correct crossing (Raviola et al., 30 Apr 2026). In a near-collapse variant with small $\theta(t)\in\mathbb{R}^p$1, standard Dirac–Frenkel eventually recovers but with an order-of-magnitude larger error than DFO. In the advection–reaction example, repeated collapse points occur at $\theta(t)\in\mathbb{R}^p$2; Dirac–Frenkel freezes at the first collapse, while DFO injects nullspace momentum at collapse and resumes accurate evolution afterwards (Raviola et al., 30 Apr 2026).

The paper also reports low- and moderate-dimensional PDE benchmarks. For rotating detonation waves on a circular domain, DFO achieves the smallest average $\theta(t)\in\mathbb{R}^p$3 error and final-time error among DF+tSVD, DF+Tikhonov, NIVP (restarts), RSNG (randomized sketches), and TENG (inner natural-gradient training), with

$\theta(t)\in\mathbb{R}^p$4

at runtime approximately equal to DF (Raviola et al., 30 Apr 2026). For 2D transport in nonuniform flow, the reported DFO errors are

$\theta(t)\in\mathbb{R}^p$5

again with smooth temporal evolution and runtime approximately equal to DF. For 2D Vlasov, DFO suppresses background error away from the solution support and yields smoother parameter velocities, with

$\theta(t)\in\mathbb{R}^p$6

In the 5D Fokker–Planck experiment for the joint density of interacting particles, DFO avoids jumps and yields smoother mean and covariance predictions. The reported mean error is $\theta(t)\in\mathbb{R}^p$7 for DFO versus $\theta(t)\in\mathbb{R}^p$8 for DF+tSVD, and the covariance error is $\theta(t)\in\mathbb{R}^p$9 for DFO versus $$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$0 for DF+tSVD. The randomized variant achieves similar accuracy at half the runtime (Raviola et al., 30 Apr 2026).

A separate inertial extension of Dirac–Frenkel dynamics reports analogous robustness patterns in different numerical settings. After semi-implicit Euler discretization, the inertial scheme solves an anchored regularized least-squares problem,

$$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$1

so that the previous velocity acts as an anchor (Raviola et al., 23 Jun 2026). In 1D Allen–Cahn and 10D Fokker–Planck experiments with neural parametrizations, that method is reported to be more robust than Tikhonov-regularized Dirac–Frenkel under strong regularization and sketching. This suggests a broader empirical pattern: history-dependent corrections are especially useful when instantaneous tangent information is weak, singular, or sample-degraded.

6. Broader mathematical context and related formulations

IRM belongs to a wider family of Galerkin-type and manifold-projection methods. In the Hilbert setting, it is the orthogonal projection of the vector field onto the tangent space of the trial manifold. In nonlinear tensor formats, this leads to dynamical low-rank evolution equations; in the Tucker fixed-rank setting on tensor Banach spaces, the fixed-rank set $$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$2 is a $$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$3-Banach manifold, the tangent space is characterized explicitly, and Dirac–Frenkel is extended through metric or generalized projections defined by the normalized duality map (Falcó et al., 2016). In Hilbert spaces, the Banach-space projection conditions reduce to the familiar orthogonality relation

$$\partial_t \hat{u}(\theta(t), \cdot) = \nabla_\theta \hat{u}(\theta(t),\cdot)^\top\,\eta(t), \qquad \eta(t) := \dot\theta(t).$$4

In quantum many-body theory, the same projection principle underlies time-dependent approximation on manifolds of quasifree states. Reformulated at the level of reduced density matrices, Dirac–Frenkel becomes instantaneous residual minimization in the Hilbert–Schmidt metric, and the corresponding tangent-space projectors yield the Bogoliubov–de-Gennes and Hartree–Fock–Bogoliubov equations (Benedikter et al., 2017). In that setting, the projected dynamics is optimal within the class of quasifree states in the sense of best instantaneous Hilbert–Schmidt residual.

Several conceptual distinctions organize the recent literature. First, IRM is a function-space statement; pathologies often arise only after pullback to redundant parameter coordinates. Second, gauge freedom is not merely a numerical nuisance but a structural property of nonlinear parametrizations with nontrivial Jacobian kernel. Third, regularization strategies are not equivalent: Tikhonov modifies the range-space component, whereas DFO preserves the IRM component exactly and changes only the nullspace representative (Raviola et al., 30 Apr 2026). Fourth, inertia-based methods and gauge-momentum methods address related defects by different mechanisms: the former yields well-posed velocity dynamics and anchored least-squares updates, while the latter imposes a nullspace-only correction that leaves the instantaneous residual-minimizing tangent dynamics unchanged (Raviola et al., 23 Jun 2026).

The resulting picture is that Dirac–Frenkel instantaneous residual minimization remains the underlying projection principle across Hilbert, Banach, tensor, neural, and operator-valued formulations. The main technical issue in nonlinear parametrizations is not the definition of the tangent-space projection but the selection of stable parameter-space representatives when the Jacobian is singular or nearly singular. The DFO principle addresses this issue by treating non-uniqueness as gauge freedom and by using minimum-dissipation-driven momentum to select temporally smooth nullspace motion without biasing the projected PDE evolution (Raviola et al., 30 Apr 2026).