LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems

• Existing approaches use iterative trajectory optimization or online guidance, including projections that can prioritize feasibility over fidelity and posterior guidance with task-dependent strengths.Source: §1 · pp. 1–2
• Feedback is inactive when pretrained dynamics provide sufficient Lyapunov decrease; otherwise, a flow-relative trust region limits corrective intervention without iterative optimization.Source: §1 · p. 2
• LyapuFlow predicts the terminal sample from each intermediate state and computes closed-form minimum-norm feedback from a prescribed Lyapunov decrease condition.Source: §1 · p. 2
• The latent-space extension propagates constraint gradients through the decoder and can restore the velocity-Jacobian contribution for finer-scale refinement.Source: §1 · p. 2
• Across physics-constrained generation in data space and image inverse problems in latent space, LyapuFlow improves sample fidelity while maintaining competitive constraint satisfaction.Source: §1 · p. 3

• The reviewed equality constraint h(x) = 0 spans physical requirements in PDEs and measurement consistency in image inverse problems.Source: §2 · p. 3

Projection-based enforcement

• Projection methods correct generated samples toward the constraint set [Hansen et al., 2023, Christopher et al., 2024], while nonlinear projection can require an m × m solve that limits scalability.Source: §2 · p. 3

Posterior and gradient-based guidance

• Posterior and task-gradient methods guide trajectories without explicit projection, but suitable guidance scales can vary widely across tasks [Chung et al., 2023, Patel et al., 2025, Kim et al., 2025, Yu et al., 2026].Source: §2 · p. 3

Control and trajectory optimization

• Trajectory-optimization methods refine complete control sequences [Liu et al., 2023, Wang et al., 2024], while robotics controllers use instantaneous Lyapunov or barrier feedback on partially denoised samples [Huang et al., 2025, Yang et al., 2025].Source: §2 · p. 3

3 LyapuFlow: Lyapunov Feedback for Constrained Generation

• LyapuFlow evaluates constraint loss on predicted terminal samples and applies minimum-norm feedback only when the pretrained flow does not meet the prescribed Lyapunov decrease.The feedback strength adapts to the current flow, constraint violation, and local gradient.Source: §3 · p. 4

3.1 Discretization and Feedback Trust Region

• The feedback trust region trades some requested Lyapunov decrease for smaller Euler discretization error by bounding the applied feedback displacement.Large finite-step corrections can move samples beyond the region where the local decrease was evaluated.Source: §3.1 · pp. 5–6

3.2 Practical Extension to Latent-Space Models

• In latent space, LyapuFlow propagates constraint gradients through the decoder and can add the velocity-Jacobian contribution later to refine high-frequency structure.Using that contribution from the start can lead to overly aggressive refinement.Source: §3.2 · p. 7
• The latent surrogate derivatives do not generally guarantee a decrease in the true Lyapunov function.In most latent-space experiments, the approximation provides sufficient constraint guidance throughout sampling.Source: §3.2 · p. 7

4 Experimental Results

• Evaluation spans physics-constrained PDE generation in data space and image inverse problems in latent space, with comparisons to representative test-time guidance baselines.The PDE benchmarks range from smooth diffusive dynamics to sharp discontinuities; image tasks use Stable Diffusion 3.5-Medium.Source: §4 · p. 7

4.1 Physics-Constrained Generation

• Across four PDE tasks, LyapuFlow has best MMSE/CMMSE throughout and best SMSE/CSMSE and FPD on three, with zero or near-zero constraint error.Burgers uses three final projections and other tasks use one; baselines retain non-negligible constraint error under the same projection budget.Source: §4.1 · p. 8, Table 1 · p. 8

4.2 Image Inverse Problems

• At matched 28 NFEs, LyapuFlow is best or second-best across image inverse-problem metrics and best in SSIM and LPIPS throughout, while preserving instance-specific details.The comparison covers ×4 super-resolution, Gaussian deblurring, and motion deblurring.Source: §4.2 · p. 9, Table 2 · p. 9

4.3 Ablation Studies

• Removing the explicit time derivative degrades fidelity on Burgers and Navier–Stokes, while removing the trust region generally degrades performance; LyapuFlow is also less sensitive to tested parameters and preserves details across NFE budgets.Compared with DPS, LyapuFlow varies less across tested decay rates; increasing FlowDPS from 28 to 40 NFEs removes some artifacts but does not recover lost instance-specific details.Source: §4.3 · pp. 10–11

5 Conclusion

• LyapuFlow evaluates constraint violations on predicted terminal samples and activates state-dependent corrections only when pretrained dynamics provide insufficient Lyapunov decrease.A feedback trust region limits control magnitude to reduce excessive finite-step corrections and resulting discretization error.Source: §5 · p. 11
• Across data and latent spaces, LyapuFlow achieves strong constraint satisfaction and high sample fidelity relative to alternative test-time constraint-enforcement strategies.The conclusion describes demonstrations across scientific machine learning and image inverse problems.Source: §5 · p. 11
• LyapuFlow requires differentiable constraint functions to construct its feedback signal, limiting direct use to constraints with available gradients.The conclusion also notes additional computational and memory overhead during generation.Source: §5 · p. 11

A Additional Analysis and Proofs

• The appendix adapts robust Lyapunov and sampled-data nonlinear-control tools to analyze the finite-step implementation, with regularity assumptions required only along the trajectories considered.The velocity field must be sufficiently smooth and the constraint loss continuously differentiable; the discrete-time result requires stronger smoothness.Source: A Additional Analysis and Proofs · p. 15

A.1 Derivation of the Minimum-Norm Lyapunov Control

• The derivation identifies the closed-form minimum-norm control as the optimal solution, with an active constraint when b > 0 and ∥g∥2 > 0.When b > 0, the constraint is active at the optimum; the derivation states t = 0 is feasible and optimal otherwise.Source: §A.1 · pp. 15–16

A.2 Discretization and Feedback Trust Region

• Finite-step feedback has a trade-off: restricting control can weaken instantaneous Lyapunov decrease, while larger control increases the second-order term.With zero shortfall, Euler integration adds O(∆t) = O(1/K) error whose constant grows with feedback magnitude; positive shortfall can leave an error contribution that grid refinement need not remove.Source: §A.2 · pp. 16–17
• The feedback trust region bounds the finite displacement caused by feedback, moderating the correction while preserving its direction; the appendix also states a sufficient condition for strict Euler-step contraction.For 0 < γα∆tk < 1 and Vk > 0, the Euler step retains strict contraction.Source: §A.2 · pp. 17–18

B Physics-Constrained Generation

• The physics-generation appendix collects PDE tasks, pretrained backbones, test-time constraints, evaluation protocols, and additional analyses.Its listed analyses include computational cost, hyperparameter sensitivity, method components, and qualitative results.Source: B Physics-Constrained Generation · p. 19

B.1.1 Tasks and Datasets

• The experiments consider four PDE benchmarks spanning different dynamics and constraint structures: 1D Heat, 1D Burgers, 1D Reaction–Diffusion (RD), and 2D Navier–Stokes (NS).Source: §B.1.1 · p. 19

B.1.2 Generative Backbone

• Unconditional functional flow-matching backbones learn distributions of complete solution fields without test-time physical conditions as explicit conditioning channels.The PDE-specific backbones use Fourier neural operators to learn velocity fields transporting a prior to the solution distribution.Source: §B.1.2 · p. 20

B.1.3 Test-Time Constraints

• At inference, constraints include direct solution conditions and conservation, balance, or local-dynamics residuals that couple parts of generated trajectories.For Burgers, the mass-balance residual accounts for boundary transport rather than requiring total mass to remain constant.Source: §B.1.3 · p. 20

B.2 Baselines and Evaluation Protocol

• The comparison includes samplers that correct intermediate states, terminal predictions, or full trajectories, and uses a common final projection after generation.The unconditional FFM reference uses that projection without an intermediate test-time correction; sampler-specific principal parameters are tuned per task.Source: §B.2 · pp. 20–21

B.3 Evaluation Metrics

• Evaluation separates physical feasibility from fidelity to the conditional solution distribution, using constraint error, conditional mean and standard-deviation errors, and Fréchet Poseidon Distance.Partial physical conditions generally leave variables unspecified and define a distribution of admissible solutions rather than a unique trajectory.Source: §B.3 · pp. 21–22

B.4 Sampling Efficiency

• LyapuFlow avoids trajectory-level iterative optimization and remains computationally competitive while applying feedback throughout sampling; OC-Flow is most expensive across the four tasks.Table 8 reports per-sample runtime and peak GPU memory including the common final projection; reported FlowDPS, ProFlow, and OC-Flow configurations use three inner iterations.Source: Table 8 · p. 22, §B.4 · p. 22

B.5 Component Ablations

• Removing the explicit time-derivative term generally increases fidelity error, especially FPD, while removing the trust region has smaller effects and leaves several metrics near the full method.The ablation interprets the time derivative as accounting for changes in predicted terminal constraint along the pretrained flow, and the trust region as protection against large finite-step corrections.Source: §B.5 · p. 23, Table 9 · p. 23

B.6 Hyperparameter Sensitivity

• Across evaluated PDE tasks, LyapuFlow varies less across tested guidance-parameter values than methods whose parameter directly controls correction step size.LyapuFlow tunes the decay rate α, which sets the requested Lyapunov decrease while feedback magnitude is determined from the current state.Source: §B.6 · p. 23, Figure 7 · p. 23

B.7 Qualitative Results

• The qualitative examples compare complete PDE solution fields under test-time conditions, including Heat, Burgers, Reaction–Diffusion, and Navier–Stokes constraint settings.The supplied captions identify the constraints shown, but do not state a qualitative winner.Source: §B.7 · p. 24, Figure 8 · p. 24, Figure 9 · p. 24

C Image Inverse Problems

• The image-inverse-problem appendix covers datasets, the pretrained backbone, degradation operators, baselines, and evaluation, then adds efficiency, larger-budget, frequency-domain, and qualitative analyses.The passage describes the section's scope rather than its findings.Source: C Image Inverse Problems · p. 28

C.1.1 Datasets and Backbone

• The evaluation uses 1k AFHQ images, 1k FFHQ images, and 0.8k DIV2K images at 768 × 768 resolution, with Stable Diffusion 3.5-Medium as the pretrained latent flow.Dataset-specific prompts and classifier-free guidance scale 2 are also specified.Source: §C.1.1 · p. 28

C.1.2 Inverse Problems and Observations

• The inverse-problem tasks are ×4 bicubic super-resolution, Gaussian deblurring, and motion deblurring with kernel size 61.Source: §C.1.2 · p. 28
• Each task applies its forward operator and adds noise with σn = 0.03; the resulting observation sets the constraint, while the same latent flow is reused across tasks.Source: §C.1.2 · p. 28

C.2 Baselines and Evaluation Protocol

• The primary comparison focuses on FlowChef and FlowDPS, while LatentDAPS, FLAIR, and FlowLPS are also included as candidate baselines.Source: §C.2 · p. 28
• Baseline configurations differ in update counts and schedules: LatentDAPS uses annealing and MCMC updates, while FlowChef, FlowDPS, FLAIR, and FlowLPS use distinct update procedures.Source: §C.2 · p. 28
• LyapuFlow’s task-specific settings for 28 and 40 NFEs are tabulated, with ksg marking when stop-gradient on vθ is turned off.Setting ksg equal to the number of NFEs keeps stop-gradient on throughout sampling.Source: §C.2 · p. 28, Table 10 · p. 29

C.3 Evaluation Metrics

• PSNR and SSIM measure reconstruction distortion and structural agreement, while LPIPS and FID capture perceptual and distributional differences.Source: §C.3 · p. 29

C.4 Sampling Efficiency

• FlowLPS improves PSNR over LyapuFlow by 1.21 dB on motion deblurring but requires more than 8× longer sampling time.This motivates the primary comparison with FlowChef and FlowDPS in a closer quality–efficiency regime.Source: §C.4 · p. 29, Table 11 · p. 29

C.5 Additional Results and Ablations

• At 40 NFEs, LyapuFlow maintains strong reconstruction quality as flow evaluations increase, following the same overall trend as the 28-NFE evaluation.Table 12 reports quantitative image-inverse-problem evaluation with Stable Diffusion 3.5-Medium at 40 NFEs.Source: §C.5 · p. 30, Table 12 · p. 30
• Intermediate gradient-switch points improve performance over full velocity-field gradients, and the best switch point varies by task and metric.Stopping velocity gradients throughout also outperforms the full-gradient setting.Source: §C.5 · p. 30, Figure 13 · p. 30

C.6 Frequency Analysis

• At 28 NFEs, spectral-density comparisons span datasets and inverse-problem tasks, distinguishing spatial-frequency recovery from aggregate reconstruction metrics.Source: §C.6 · p. 31, Figure 14 · p. 31

C.7 Qualitative Results

• AFHQ qualitative comparisons cover bicubic super-resolution, Gaussian deblurring, and motion deblurring with Stable Diffusion 3.5-Medium at 28 NFEs.Source: Figure 15 · p. 32, §C.7 · p. 32, Figure 16 · p. 33
• FFHQ qualitative comparisons cover bicubic super-resolution, Gaussian deblurring, and motion deblurring with Stable Diffusion 3.5-Medium at 28 NFEs.Source: §C.7 · p. 32, Figure 18 · p. 35, Figure 19 · p. 36
• DIV2K qualitative comparisons cover bicubic super-resolution, Gaussian deblurring, and motion deblurring with Stable Diffusion 3.5-Medium at 28 NFEs.Source: §C.7 · p. 32, Figure 21 · p. 38, Figure 22 · p. 39