Solver-Trajectory Flow Matching

Integrate this at the training and inference interfaces of a conditional flow or diffusion model, using trajectory data from any expensive iterative teacher. The tensor being modified is the model state $x$ itself: an image latent, token embedding sequence, graph state, or simulator design vector. First, for every training target $y_K$, run or retrieve a teacher trajectory $(y_0,y_1,\ldots,y_K)$, where $y_0$ is Gaussian noise and later states are optimiser or refinement iterates. Normalize iteration indices to $s_k=k/K$, or use actual cumulative step lengths if teacher steps have unequal magnitudes. At each minibatch, sample a segment index $k$ uniformly and sample $r\sim\mathrm{Uniform}(0,1)$. Set $t=(1-r)s_k+rs_{k+1}$, compute $x_t=(1-r)y_k+ry_{k+1}$ and $u_t=(y_{k+1}-y_k)/(s_{k+1}-s_k)$, then backpropagate $w_k\|v_\theta(x_t,t,c)-u_t\|_2^2$. For a first implementation set $q_k=1$ and compare $\lambda=0,0.25,1,4$; alternatively define $q_k$ from the teacher's per-step objective decrease and normalize it to $[0,1]$. At inference, start from Gaussian noise and integrate the learned ODE with Euler or Heun using 4, 8, 16, and 32 steps. The exact mathematical computations are the interpolation point, segment velocity, weighting rule, and ODE update; empirical quantities are the teacher anchors and confidence scores. Begin with a small conditional MNIST or CIFAR-10 experiment, using a U-Net or MLP flow model and standard endpoint flow matching as the baseline. Measure FID or validation loss at equal numbers of function evaluations, diversity, and distance to the teacher's final state. For optimisation data, also measure constraint violation and objective value. Success is lower error or better FID at 4–8 steps without sample collapse, plus improved feasibility and teacher-objective quality.