Closed-Form DynaBase Calibration
Implementation & benchmark of arXiv:2607.14937 — A Minimal Interpretable Architecture for Zero-Shot Reconstruction of Dynamical Systems
Source paper: A Minimal Interpretable Architecture for Zero-Shot Reconstruction of Dynamical Systems arXiv:2607.14937 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Calibrate the two blend coefficients directly from a context trajectory rather than using gradient descent. The one-step prediction problem is a two-variable ridge regression, making per-task adaptation nearly free and suitable for zero-shot or few-shot system identification.
Formulas
Mathematical statement
For each demonstration transition, let $\bm{q}_{r}$ be the current latent state, $\bm{n}_{r}$ its nearest retrieved context state, $\bm{s}_{r}$ that state's temporal successor, and $\bm{y}_{r}$ the observed next latent state. The adapted DynaBase residual equation is $\bm{y}_{r}-\bm{q}_{r}=\alpha(\bm{n}_{r}-\bm{q}_{r})+\beta(\bm{s}_{r}-\bm{q}_{r})$. Stack these equations over $r=1,\ldots,R$: define $\bm{X}\in\mathbb{R}^{RM\times2}$ by placing the two $M$-vectors $\bm{n}_{r}-\bm{q}_{r}$ and $\bm{s}_{r}-\bm{q}_{r}$ in row-block $r$, and define $\bm{y}\in\mathbb{R}^{RM}$ by stacking $\bm{y}_{r}-\bm{q}_{r}$. The ridge solution $\theta=(\alpha,\beta)^{\top}=(\bm{X}^{\top}\bm{X}+\lambda I_{2})^{-1}\bm{X}^{\top}\bm{y}$ is the exact minimizer of squared one-step error plus $\lambda\|\theta\|_{2}^{2}$. If convex blending is required, project $(\alpha,\beta)$ onto $\{(a,b):a\geq0,b\geq0,a+b\leq1\}$ after solving, or solve the corresponding two-variable constrained quadratic program.
Implementation notes
(1) Integration point: use this as the adaptation or calibration step for the retrieval forecast head, before rolling out a query trajectory. It can also replace the final coefficient layer of a latent world model during test-time system identification. (2) Pseudocode: Q=encode(context[:-1]); Y=encode(context[1:]); for r in range(R): i=NN(Q[r], Q[:R-1]); X[Mr:M(r+1),:]=[Q[i]-Q[r], Y[i]-Q[r]]; y[Mr:M(r+1)]=Y[i]-Q[r]; theta=solve(X.T@X+lambda*I, X.T@y); theta=project_simplex(theta); rollout using z_next=(1-a-b)*z+a*C[i]+b*S[i]. In the actual calibration data, use the observed next state $Y[r]$ as the target, so the row-block target is Y[r]-Q[r], not the retrieved successor. (3) The closed-form matrix solve is computed exactly; nearest-neighbor assignments, latent representations, ridge strength $\lambda$, and whether coefficients are shared across tasks are empirical choices. Fit on the first half of each context and evaluate on held-out transitions to avoid leakage. (4) First cheap experiment: use a 2-layer MLP encoder with 16-dimensional latents on 1,000 short Lorenz-63 and Van der Pol trajectories. Compare closed-form calibration against Adam optimization of the same two coefficients for equal wall-clock adaptation budgets, plus fixed coefficients. Success is lower held-out one-step MSE and lower 32-step rollout error with under 1 ms coefficient fitting per task; an additional useful signal is that calibration reaches its final error in one pass rather than requiring dozens of gradient steps.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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