Response Renormalization for Critical Deep Equilibrium Models
Authors: Jose Luis Lima de Jesus Silva
Organizations: Federal University of Bahia, Department of Geophysics Salvador, BA 40170-115, Brazil · Grupo de Estudos e Aplicação de Inteligência Artificial em Geofísica (GAIA) Federal University of Bahia, Salvador, BA 40170-115, Brazil
Deep Equilibrium Models (DEQs) compute predictions from a hidden representation unchanged by the model update. Training through this equilibrium uses implicit differentiation and requires solving an adjoint system built from the residual Jacobian. If this Jacobian is nearly singular along loss-sensitive directions, small perturbations can be strongly amplified in the adjoint response, producing large, highly sensitive gradients that can make optimization unreliable. We introduce Response Renormalization, a backward-pass framework that lifts selected near-pole denominators while leaving unlifted response channels unchanged. Collective Mode Response Renormalization (CMR) applies this correction in a low-dimensional critical subspace, while Phi-adaptive CMR computes a bounded response mass from a positive susceptibility rule. We derive dense and matrix-free collective formulations, distinguish exact gradients of a modified frozen-anchor residual from backward-response surrogates, and extend the construction to Structured Implicit Layers and Vector Attractors (SILVA). Across 23 multiphysics families spanning partial differential equations, three-dimensional fields, operator maps, complex geometries, and particle systems, CMR and Phi-CMR yield test errors no more than five percent higher than those from models trained with exact implicit differentiation in more than 98% of static and 95% of transient family-seed comparisons. Solver-index experiments show convergence toward the static adjoint, while physical-time rollouts retain predictive fidelity under the evaluated conditions. These results demonstrate that selective response renormalization can control near-critical adjoint amplification without globally damping well-conditioned sensitivity. Therefore, the method can make parameter updates more reliable while preserving the useful gradient information needed for learning.
Figures & tables
Figure 1: Pole-response maps with (a) dense pole strength −log10σmin , (b) the Phi-CMR mass multiplier, (c) RΦ−CMR−RTik at critical rank one, and (d) the same difference at critical rank two over target gap and state dimension.
Figure 2: Selective renormalization of the near-pole response with (a) exact and filtered response amplification versus singular value, (b) original and renormalized denominators for stable and critical modes, (c) Phi-CMR mass versus pole pressure by state dimension and critical rank, and (d) retained response versus residual against the original adjoint equation.
Block
Method
n
Rm
Cm
Pm
Em
Collective grid
Phi-CMR
36
0.303 ± 0.239
0.947 ± 0.061
0.290 ± 0.234
0.726 ± 0.202
Collective grid
Tikhonov
36
0.231 ± 0.250
0.815 ± 0.191
0.206 ± 0.245
0.812 ± 0.208
Source rotation
CMR
48
0.471 ± 0.283
0.999 ± 0.001
0.471 ± 0.283
0.530 ± 0.283
Source rotation
Tikhonov
48
0.242 ± 0.307
0.973 ± 0.055
0.241 ± 0.308
0.760 ± 0.308
Held-out gate
Delta-Phi
20
0.440 ± 0.351
0.999 ± 0.002
0.440 ± 0.351
0.560 ± 0.351
Held-out gate
Phi-CMR
20
0.432 ± 0.350
0.997 ± 0.005
0.432 ± 0.350
0.569 ± 0.350
Table 1: Directional dense-adjoint evidence. Entries are mean ± standard deviation over configurations. Rm is the norm ratio, Cm is directional cosine, Pm is signed projection onto the exact adjoint, and Em is relative adjoint error.
Figure 3: Darcy field bridge with (a) target field, (b) Phi-CMR absolute error, (c) Tikhonov-minus-Phi-CMR absolute-error difference, (d) source-weighted pole mode, (e) coefficient field with solution contours, (f) Delta-Phi absolute error with pole contours.
Method
One-step error
Ratio ≤1.05
Time
Horizon-8 error
Ratio ≤1.05
Implicit
1.000
50/50
1.00
1.000
45/45
Phantom
0.988
48/50
0.65
0.997
43/45
Neumann
1.015
43/50
0.58
1.007
42/45
JFB
1.011
43/50
0.52
0.988
42/45
SHINE
1.060
35/50
0.55
1.052
31/45
Tikhonov
1.011
47/50
2.49
1.016
43/45
Table 2: All native SILVA baselines over ten one-step families and nine dynamic families. Errors and time are normalized by exact implicit differentiation.
Appendix figures & tables47 assets
Supplementary material from the paper’s appendix.
Appendix
Native dataset
GB
Cross-benchmark dataset
GB
PDEBench Darcy 2D (five regimes)
6.554
PDEArena Navier–Stokes 2D
0.826
PDEBench Advection 1D
8.233
PDEArena Shallow Water 2D
0.089
PDEBench Burgers 1D
8.233
PDEArena Maxwell 3D
1.887
PDEBench Reaction–Diffusion 1D
4.137
PDEArena Kuramoto–Sivashinsky 1D
0.931
PDEBench Diffusion–Sorption 1D
4.217
DynaBench Advection
2.558
PDEBench Diffusion–Reaction 2D
13.244
DynaBench Burgers
5.090
Appendix
Table 3: Storage footprints of the selected data subsets. Values are decimal gigabytes ( 1GB=109 bytes) and exclude unused portions of the complete source collections.
Figure 4: Native SILVA transient analysis with (a) family-weighted free physical-time rollout error relative to models trained with exact implicit differentiation on nine dynamic datasets and five seeds, (b) finite causal-response error relative to the converged static adjoint, and (c) fixed-point state error relative to equilibrium, where panels (b,c) contain 45 dataset–seed models per method. Shading in panel (a) denotes error ratios at most 1.05.
Diagnostic
Observed
Reference level
Maximum final forward residual
9.98×10−8
<10−5
Maximum implicit/CMR/Phi adjoint residual
9.99×10−9
<10−6
Structured-method realizations improving from initialization
57.0%
Descriptive
Families with collective activation
10/10
10/10
Phi-CMR error ratio ≤1.05
48/50 pairs
40/50 pairs
Phi-CMR lower error than Tikhonov
27/50 pairs
25/50 pairs
Appendix
Table 4: SILVA convergence, fidelity, and activation diagnostics. Statistics use physical-family weighting, with the five Darcy regimes contributing one family-level unit.
Figure 5: Native SILVA multiphysics evaluation with (a) final inverse-standardized sample-relative Euclidean error normalized by exact implicit differentiation and paired-seed 95% intervals, (b) paired Phi-CMR-minus-Tikhonov differences, (c) the fraction of training updates with a lifted collective response, and (d) measured forward-solve plus backward wall time relative to implicit differentiation, with the Darcy regimes aggregated as one physical family. Shading in panel (a) denotes error ratios at most 1.05. The Darcy confidence interval in (a) extends beyond the displayed range.
70 and 65 family/regime–seed model units, 1,080 method evaluations
Five seeds, eight backward modes, disjoint dataset-provided splits, identical forward models and ordered minibatches within every paired comparison.
Capacity extension
50 exploratory aligned settings and 400 evaluations, plus 40 transient-aligned evaluations
Six high-error families, hidden widths 96 or 128, resolutions up to 642 or 163 , and the same eight backward modes.
Transient evaluation
2,880 physical-time and 945 solver-time repeated evaluations
Free autoregressive rollouts without retraining and solver horizons 1,2,4,8,16,32,64 , compared with converged equilibria and GMRES adjoints.
Appendix
Table 6: Evaluation blocks and statistical units.
Block
Metric
n
A
B
A–B
Phi-CMR vs Tikhonov
response-norm ratio
36
0.303
0.231
0.073
CMR vs Tikhonov, source sweep
response-norm ratio
48
0.471
0.242
0.229
High vs low source alignment
CMR gain over Tikhonov
36
0.263
0.172
0.091
Phi-CMR vs Tikhonov, Darcy
relative field error
1
1.211
1.558
-0.347
Delta-Phi vs Phi-CMR
low-source norm ratio
20
0.668
0.623
0.045
Delta-Phi vs Phi-CMR
high-source suppression
20
0.816
0.758
0.057
Appendix
Table 7: Extended quantitative evidence. Negative field-error differences favor the first method.
Figure 6: Rank-two response with (a) pole strength and (b) the rank-two minus rank-one Phi-CMR advantage over Tikhonov.
Figure 7: Source-alignment experiment with (a) retention gain over Tikhonov and (b) response norm versus critical-mode suppression.
Figure 8: Delta-Phi sweep with (a) bounded mass, (b) strength utility, and (c) response norm versus critical-mode suppression.
Figure 9: Delta-Phi ablation with (a) gate activation, (b) mass correction, and (c) suppression gain.
Source
Dataset
Physical setting
Representation
PDEArena
Navier–Stokes 2D
incompressible flow
multi-field grid
PDEArena
Shallow Water 2D
free-surface flow
multi-field grid
PDEArena
Maxwell 3D
electromagnetic waves
vector-field volume
PDEArena
Kuramoto–Sivashinsky 1D
chaotic dissipative dynamics
scalar grid
DynaBench
Advection
linear transport
structured grid
DynaBench
Burgers
nonlinear conservation law
structured grid
Appendix
Table 8: Cross-benchmark evaluation. Dynamic entries use one-step training and seven-step free physical-time rollouts, while Poisson–Gauss is evaluated as a static source-to-solution operator.
Method
Static error
Ratio ≤1.05
Time
Horizon-7 error
Ratio ≤1.05
Implicit
1.0000
65/65
1.00
1.0000
60/60
Phantom
0.9994
64/65
0.59
0.9969
60/60
Neumann
0.9975
64/65
0.54
0.9922
60/60
JFB
0.9982
63/65
0.48
0.9962
60/60
SHINE
0.9818
62/65
0.52
0.9922
55/60
Tikhonov
0.9989
65/65
2.48
0.9987
58/60
Appendix
Table 9: All-method cross-benchmark results. Static values use 65 family–seed pairs, while horizon seven uses 60 pairs from the 12 dynamic datasets. Error and measured solve-plus-backward time are normalized by exact implicit differentiation.
Figure 10: Cross-benchmark evaluation of SILVA response renormalization with (a) final inverse-standardized sample-relative Euclidean error normalized by exact implicit differentiation, (b) paired Phi-CMR-minus-Tikhonov differences across seeds, (c) the fraction of training measurements with an active collective lift, and (d) measured equilibrium-solve plus backward time relative to implicit differentiation. Shading in panel (a) denotes error ratios at most 1.05.
Figure 11: Cross-benchmark transient evidence with (a) seven-step free physical-time rollouts for all eight backward methods on 12 dynamic datasets, (b) finite causal-response error relative to the static implicit adjoint, and (c) original DEQ state error relative to equilibrium, where panels (b,c) use implicit, CMR, and Phi-CMR models at solver horizons N∈{1,2,4,8,16,32,64} . Shading in panel (a) denotes error ratios at most 1.05.
Family
Baseline
Capacity
Ratio
Paired delta 95% CI
Seeds improved
Darcy 2D
1.1119
0.6483
0.583
[−0.6957,−0.2314]
5/5
Diffusion–Reaction 2D
1.0170
0.9931
0.977
[−0.0258,−0.0219]
5/5
PDEArena Navier–Stokes
0.6405
0.4496
0.702
[−0.1951,−0.1866]
5/5
PDEArena Shallow Water
0.3629
0.3287
0.906
[−0.0536,−0.0147]
5/5
PDEArena Maxwell 3D
0.9674
0.6244
0.645
[−0.3588,−0.3272]
5/5
LagrangeBench TGV
0.2204
0.1252
0.573
[−0.1235,−0.0668]
5/5
Appendix
Table 10: Phi-CMR inverse-standardized sample-relative Euclidean error in the forward-capacity extension, with capacity-to-baseline ratios and paired 95% intervals for the capacity-minus-baseline difference.
Figure 12: Held-out prediction geometry over five seeds with (a) representative-field relative error, (b) centered spatial correlation, and (c) prediction-to-target amplitude ratio, where points are individual seeds, short horizontal marks are means, and Diffusion–Reaction uses the separate t1→t2 transient task.
Figure 13: Higher-capacity Darcy 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Panels (a–e) use the median-error β=1 example, while panel (f) aggregates all five Darcy regimes.
Figure 14: Transient-aligned Diffusion–Reaction 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first physical channel for the separately evaluated t1→t2 task.
Figure 15: Higher-capacity PDEArena Navier–Stokes 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first vorticity/state channel.
Figure 16: Higher-capacity PDEArena Shallow Water 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first pressure/height channel.
Figure 17: Higher-capacity PDEArena Maxwell 3D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels show the central slice of the first electric-displacement component after vector-aware Conv3D reconstruction.
Figure 18: Higher-capacity LagrangeBench Taylor–Green vortex 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Panels (a–e) retain the native particle coordinates.
Figure 19: PDEArena Navier–Stokes 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first vorticity/state channel.
Figure 20: PDEArena Shallow Water 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first pressure/height channel.
Figure 21: PDEArena Maxwell 3D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels show the central slice of the first electric-displacement component.
Figure 22: PDEArena Kuramoto–Sivashinsky 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the scalar state.
Figure 23: DynaBench Advection with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the advected scalar.
Figure 24: DynaBench Burgers with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the conserved scalar.
Figure 25: DynaBench Gas Dynamics with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display density, while panel (f) aggregates all predicted variables.
Figure 26: DynaBench Kuramoto–Sivashinsky with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the scalar state for the deterministic example nearest the Phi-CMR median error (offset 3 of 16).
Figure 27: DynaBench Reaction–Diffusion with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first reaction–diffusion channel.
Figure 28: DynaBench Wave with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first wave-state channel.
Figure 29: PDEGym Poisson–Gauss with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Panels (a–e) show the source-to-solution operator example.
Figure 30: CFDBench cylinder geometry with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display velocity for a held-out geometry case.
Figure 31: LagrangeBench Taylor–Green vortex 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Panels (a–e) retain the native particle coordinates, and panel (d) is position error.
Physical family
Mathematical setting
Displayed quantity
Advection 1D
linear transport
state
Burgers 1D
nonlinear conservation law
state
Reaction–Diffusion 1D
coupled reaction and diffusion
state
Diffusion–Sorption 1D
reactive transport
concentration
Diffusion–Reaction 2D
coupled parabolic system
first field
Shallow Water 2D
hyperbolic free-surface flow
height
Appendix
Table 11: Physical systems and displayed quantities in the native SILVA per-family analysis. Darcy is evaluated as an elliptic coefficient-to-solution map, while the remaining entries are one-step dynamical predictions.
Figure 32: Advection 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the scalar state.
Figure 33: Burgers 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the scalar state.
Figure 34: Reaction–Diffusion 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first state channel.
Figure 35: Diffusion–Sorption 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display concentration.
Figure 36: Diffusion–Reaction 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first physical channel.
Figure 37: Shallow Water 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display fluid height.
Figure 38: Compressible CFD 1D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display density, while panel (f) aggregates density, pressure, and velocity.
Figure 39: Incompressible Navier–Stokes 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display horizontal velocity.
Figure 40: The Well turbulent radiative layer 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Field panels display the first physical channel from the held-out split.
Figure 41: Darcy 2D with (a) input, (b) target, (c) Phi-CMR prediction, (d) absolute error, (e) Tikhonov-minus-Phi-CMR error, (f) five-seed all-method relative- L2 , (g) original and effective collective denominators with active lifts, and (h) source-weighted singular response. Panels (a–e) use a fixed β=1 coefficient-to-solution example, while panel (f) aggregates five Darcy regimes.
Deep equilibrium models promise input-adaptive implicit computation: harder problems should demand more solver iterations, and the solved equilibrium should encode the result of genuine iterative inference. We report a cautionary study of a port-Hamiltonian DEQ with a learned initialization on two reasoning tasks -- ProofWriter entailment over frozen DeBERTa embeddings and a BFS-verified graph-reachability benchmark -- in which the implicit computation is a silent no-op. Across tasks, seeds, and controlled ablation arms, the solved equilibrium equals the solver's start point to numerical precision, and bypassing the solver entirely changes test accuracy by +0.00 percentage points in 18 of 19 training runs. Controlled interventions falsify the tempting explanation: removing the anchoring term reproduces every result, and retraining with noise-decoupled starts yields a solver that converges to the noisy start while the decoder learns to ignore it. The single escaping run diverges instead (∥h∗−z0∥=171), producing a co-adapted noise channel whose removal improves accuracy. Iteration counts are uncorrelated with ground-truth difficulty (r=0.009), and the full apparatus never outperforms a two-layer MLP on either task. We trace the mechanism to gradient starvation along two distinct routes, show that the standard zeroing ablation is confounded and gives wildly seed-dependent answers where the correct substitution test gives a stable zero, and distill a four-test diagnostic protocol for auditing claimed implicit computation. All experiments run on a single free Colab GPU; code, raw logs, and analysis scripts are released.
In situ adjoint training extracts parameter gradients directly from measurement, but has so far been limited to reciprocal or restricted systems. Here, we introduce the physical counterpart of the policy gradient theorem: a stochastic-adjoint gradient estimator that lifts these constraints by trading reciprocity for nondegenerate diffusion. As validation, we train a nonlinear resonator network, whose own dynamics supply the policy, against antagonistic temporal modulations with gradients from measured stochastic trajectories alone, without finite differences or a separate adjoint experiment.
William Tuxbury, Zin Lin
Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
Training-free diffusion priors enable inverse-problem solvers without retraining, but for nonlinear forward operators data consistency often relies on repeated derivatives or inner optimization/MCMC loops with conservative step sizes, incurring many iterations and denoiser/score evaluations. We propose a training-free solver that replaces these inner loops with a hard measurement-space feasibility constraint (closed-form projection) and an analytic, model-optimal step size, enabling a small, fixed compute budget per noise level. Anchored at the denoiser prediction, the correction is approximated via an adjoint-free, ADMM-style splitting with projection and a few steepest-descent updates, using one VJP and either one JVP or a forward-difference probe, followed by backtracking and decoupled re-annealing. We prove local model optimality and descent under backtracking for the step-size rule, and derive an explicit KL bound for mode-substitution re-annealing under a local Gaussian conditional surrogate. We also develop a latent variant and a one-parameter pixel→latent hybrid schedule. Experiments achieve competitive PSNR/SSIM/LPIPS with up to 19.5× speedup, without hand-coded adjoints or inner MCMC.
Minwoo Kim, Seunghyeok Shin, Hongki Lim
Department of Electrical and Computer Engineering, Inha University Incheon, 22212, South Korea