FlowSteer: Trajectory-Guided Inference & Planning

• FlowSteer is a family of analytic and algorithmic approaches for trajectory-guided inference and planning, blending measurement-aware corrections with learned flow priors.
• In image restoration and synthesis, FlowSteer employs innovative scheduler architectures with windowed lambda weights and null-space projections to enhance fidelity and perceptual quality.
• In robotics and free-surface modeling, FlowSteer integrates analytic distance functions and coordinate transforms to yield efficient, physically consistent motion planning.

FlowSteer (FS) denotes a family of analytic and algorithmic approaches engineered for trajectory-guided inference and planning across multiple domains, including image restoration, few-step visual synthesis, and robotics in flow fields. The paradigm centers on controller- or measurement-aware scheduling of corrections along a latent continuous trajectory, leveraging structural and measurement constraints to maximize fidelity, perceptual consistency, and computational efficiency. FS is also used for free-surface modeling in hydrostatic incompressible flows in closed pipes, arising from formal asymptotic reductions of the 3D Euler equations.

1. FlowSteer in Operator-Aware Conditioning for Image Restoration

FlowSteer, as introduced for image restoration in flow-based models, is a scheduler-based conditioning scheme that couples a frozen generative flow field (e.g., regression flow matching—RF models) to explicit measurement priors via proximal projections along the ODE sampling path. The formulation is designed for zero-shot enhancement of measurement fidelity—injecting measurement consistency into inference without model retraining, adapters, or extra networks (Wickremasinghe et al., 9 Dec 2025).

• Let $x_t = t x_0 + (1-t)x_1$ be the trajectory between data $x_0$ and noise $x_1$, where $v_\theta(x_t, t; C)$ is the learned velocity field (prompt-conditioned via $C$), and $z_t$ is the latent VAE code.
• At each timestep, the update alternates between a flow-driven denoising step (applying velocity field along the learned ODE) and a measurement-consistency step via null-space projection:

$$x_{t_i} \leftarrow A^\dagger y + \lambda_i (I - A^\dagger A)x_{t_i}$$

where $A$ is the measurement operator (e.g., downsampling, convolution), $A^\dagger$ is its pseudo-inverse, and $\lambda_i$ is a scheduler weight.

• The scheduler applies nontrivial per-timestep weighting (e.g., windowed $\lambda_i$ from $i_{\text{start}}\approx0.5N$ to $i_{\text{end}}\approx0.9N$), avoiding early noise injection and late hallucination.

The method demonstrates superior PSNR, SSIM, LPIPS, and CLIP-cosine metrics over baselines across colorization, super-resolution, deblurring, and denoising. Key findings include significant gains via two-step $\lambda_i$ scheduling and decode–encode strategies, avoiding noise accumulation observed in projection-to-$x_0|t$ strategies (Wickremasinghe et al., 9 Dec 2025).

Task	Method	PSNR	SSIM	LPIPS	CLIP
Colorization	FlowSteer	27.42	0.870	0.208	0.773
Super-res	FlowSteer	32.86	0.902	0.170	0.671
Deblurring	FlowSteer	32.87	0.905	0.149	0.818
Denoising	FlowSteer	32.21	0.892	0.182	0.768

2. FS in Few-Step Image Synthesis: Trajectory-Guided Distillation

In few-step synthesis, FlowSteer is implemented to mitigate performance bottlenecks in ReFlow-based distillation (PeRFlow, InstaFlow) for visual generation models. The FS method addresses two critical distribution mismatches: (1) “teacher trajectory mismatch,” where training relies on off-trajectory start points instead of authentic teacher ODE states, and (2) “inter-stage mismatch,” resulting from segment transitions at inference based on previous ODE solutions rather than fresh interpolations (Ke et al., 24 Nov 2025).

• Online Trajectory Alignment (OTA) realigns training by sampling teacher-generated states at stage transitions, enforcing true on-policy velocity matching:

$v^* = \frac{z_{t_{k-1}} - z_{t_k}}{t_{k-1} - t_k}$

where $z_{t_k}$ is obtained via the teacher's ODE solve.

• Adversarial distillation introduces a discriminator on trajectory states $(z, t)$, jointly optimized with a feature-matching loss:

$$\mathcal{L}_{\text{adv}} = - \mathbb{E}[D(z_t^S, t)], \quad \mathcal{L}_{\text{FM}} = \sum_l \|D_l(z_t^T, t) - D_l(z_t^S, t)\|_2$$

culminating in a weighted sum with OTA's velocity alignment loss.

• A correction to the FlowMatchEulerDiscreteScheduler ensures proportional step sizes in N-step inference, preventing instability and fidelity degradation in the final step.

Experimental results on SD3 and SD3.5-Large demonstrate state-of-the-art PickScore and HPSv2 metrics for 4-step synthesis, outperforming PCM/Hyper-SD and Flash Diffusion distillation. OTA and scheduler corrections yield the largest gains, with adversarial trajectory distillation further enhancing text-image faithfulness and artifact suppression (Ke et al., 24 Nov 2025).

3. Streamline-Based Flow Field Planning and Analytic Metrics

In robotics and motion planning under strong incompressible flows, FlowSteer introduces pseudo-metrics that embed the physical structure of the flow field into planning heuristics (To et al., 2020):

• The L²-stream distance augments Euclidean distance with scaled stream function difference:

$$d_{\text{stream}}(x_P, x_Q) = \sqrt{\|x_P - x_Q\|_2^2 + \left[\psi(x_P, x_Q)/\alpha\right]^2}$$

with stream function $$\psi(x_P, x_Q) = \int_{x_P}^{x_Q} [u_n(x) dy - v_n(x) dx]$$ and scaling $\alpha$.

• The L²-LSB distance incorporates the lower speed bound (minimum speed needed to cross an adversarial flow):

$$V_{\text{LSB}}(x_P,x_Q) = \frac{|\psi(x_P,x_Q)|}{\|x_Q - x_P\|_2}$$

$$d_{\text{LSB}}(x_P,x_Q) = \sqrt{\|x_P - x_Q\|_2^2 + [V_{\text{LSB}}\cdot \beta]^2}$$

where $\beta$ is a characteristic time.

• Steering heuristics enforce the streamline-crossing constraint in control space:

$\psi(x_P, x_Q) + u_s \Delta y - v_s \Delta x = 0$

intersecting the feasible control line with the control magnitude constraint circle to select progress-maximizing trajectories.

Integrated into RRT* frameworks, these metrics and control heuristics yield faster and higher-quality path planning in simulations and field data, notably outperforming vanilla and VF-RRT* algorithms in artificial vortices and ocean current benchmarks, with analytic nearest-neighbor indexing and adaptive arc-length integration suitable for embedded platforms (To et al., 2020).

4. Free-Surface Modeling: FS-Model for Unsteady Pipe Flows

FS also refers to the Free-Surface model for unsteady, incompressible flows in closed, non-uniform pipes. The derivation proceeds from the 3D Euler equations to a 1D shallow-water system via a Serret-Frenet frame (local to pipe centerline) and a thin-layer $\epsilon \to 0$ expansion (Bourdarias et al., 2011):

• Variables include curvilinear abscissa $X$, wetted area $A$, discharge $Q = A \overline{u}$, free-surface elevation $h$, and geometric functions $\sigma(X,z)$ for local width.
• The FS-model equations:

$\partial_t A + \partial_X Q = 0$

$$\partial_t Q + \partial_X \left[\frac{Q^2}{A} + g I_1(X, A) \cos \theta(X)\right] = -g A \sin \theta(X) + g I_2(X, A) \cos \theta(X) - g A \overline{z}(X, A) \partial_X[\cos \theta(X)]$$

with hydrostatic pressure integral $I_1$, source $I_2$, and centroid coordinate $\overline{z}$.

• Boundary conditions enforce no-leak at walls and kinematic free-surface, with atmospheric surface pressure.

Regimes of validity include slender-flow ($\epsilon = H/L \ll 1$), hydrostatic core, inviscid flow with phenomenological friction, and moderate curvature/slope (Bourdarias et al., 2011).

5. Design Principles and Scheduler Architectures

Across all domains, FlowSteer is characterized by principled scheduler architectures that balance the injection of measurement or control corrections with the propagation of learned (or physical) flow priors:

• In image restoration and synthesis, windowed or multi-step $\lambda_i$ schedules are essential to stability and fidelity, preventing unaccounted noise amplification and hallucination.
• In robotics, analytic distance functions and adaptive step sizes allow flow-aware exploration while preserving computational efficiency and observability.
• In free-surface hydrostatics, dimensional reductions and coordinate transforms ensure physical consistency, and regularizations ensure conservative, hyperbolic dynamics under geometric variations.

A plausible implication is that suitably engineered FlowSteer routines generalize beyond their founding domains, provided the underlying trajectory and update operators are compatible with the analytic or learned flow models.

6. Practical Implications, Limitations, and Experimental Outcomes

FlowSteer implementations exhibit notable strengths in zero-shot capability (frozen priors, no retraining), measurement/topology-aware corrections, and efficiency in both inference and planning contexts:

• In vision, FS achieves simultaneous measurement fidelity and perceptual sharpness with strict zero-shot deployment (Wickremasinghe et al., 9 Dec 2025).
• In planning, FS yields faster first feasible and lower-cost final solutions in strong flow fields, supporting embedded system deployment (To et al., 2020).
• Limitations include sensitivity to scheduler tuning in vision tasks (window positions, step sizes), and assumed stationarity (flow field invariance) in planning metrics.
• In free-surface modeling, assumptions on pipe rigidity, slenderness, and curvature impose bounds on model fidelity to real-world pipe dynamics (Bourdarias et al., 2011).

Collectively, FS methodologies demonstrate the efficacy of guided sampling and control under explicit process, measurement, and physical constraints, with broad applicability across inference, synthesis, and dynamical modeling.