perturbed initial noisy defined score

The gap

Supporting evidence — 2 representative gaps

• FRMD: fast robot motion diffusion via trajectory-level consistency distillation (2026) · doi

  FRMD (ours) Action-Space Diffusion (Janner et al., 2022; Chi et al., 2023; Black et al., 2024) High-dimensional action space (nk); 10–100 denoising steps; latency > 100 ms Low-dimensional MP space (d ≪ nk); single-step inference; 17 ms runtime (∼7 × speedup) Movement Primitives (Ijspeert et al., 2013; Paraschos et al., 2013; Doe and Smith, 2017) Deterministic or hand-tuned; limited expressiveness for complex distributions Learns stochastic diffusion over MPs weights; scalable to multi-modal behaviors Consistency Models (Song et al., 2023; Kim et al., 2023; Dai et al., 2024) Limited to vision/language domains; no robotic control validation Trajectory-level consistency distillation for robotics Trajectory-Parameterized Diffusion (Carvalho et al., 2024; Scheikl et al., 2024) Multi-step denoising remains; not real-time feasible One-step consistency mapping; achieves real-time control with diffusion (∼10 × speedup) FRMD, combines trajectory-space diffusion with consistency distillation to achieve real-time inference while maintaining task success. ℝk×d. For convenience, we vectorize this matrix as w ∈ ℝkd in the following sections. Decoder Mapping. The mapping from ProDMP weights to a discrete action trajectory is defined as: τ = G (w; y0, ̇y0) , (5) where G denotes the deterministic ProDMP decoder that evaluates Equations 3, 4 across time steps and degrees of freedom to produce the full action sequence τ ∈ ℝn×k. 3.2.2 Consistency models Consistency models (Song et al., 2023) are an alternative to standard diffusion models that aim to reduce inference cost by learning direct mappings from noisy inputs to clean data. Instead of generating samples through iterative denoising, a consistency model predicts the final output in a single forward pass conditioned on the noise level. Probability Flow ODE. Diffusion models can be formulated as solving a continuous-time ordinary differential equation (ODE), known as the probability flow ODE (PF-ODE). Given a data sample x0 ∼ pdata, the forward noising process is defined as Equation 6: xs = x0 + σ (s) ϵ, ϵ ∼ N (0, I) , (6) where s ∈ [0, S] denotes diffusion time (distinct from trajectory time t), and σ(s) is a monotonically increasing noise schedule with σ(0) = 0 and σ(S) = σmax. The corresponding reverse process is governed by the PF-ODE as formalized in Equation 7: dx ds = − ̇σ (s) σ (s) ∇x log ps (x) , (7) where ∇x log ps(x) is the score function of the perturbed data distribution. Numerically integrating this ODE from s = S to s = 0 recovers a clean sample from an initial Gaussian noise input. Consistency Function. A consistency model parameterizes a function fθ(xs, s) that predicts the corresponding clean sample x0 from any noisy state xs along the PF-ODE trajectory. The defining property is self-consistency as showed in Equation 8: predictions from different diffusion times

  Keywords: consistency diffusion time trajectory models action space equation denoising step inference real mapping clean noise
• An efficient higher-order trigonometric cubic B-spline collocation method for timefractional Burgers equations (2026) · doi

  Robustness testing with perturbed initial/boundary conditions or noisy data is not discussed.

  Keywords: robustness testing perturbed initial boundary conditions noisy discussed