# Closure-Decorrelation Memory Scheduler - ID: 2504 - Canonical URL: https://synthcore.org/idea/2504/closure-decorrelation-memory-scheduler - API JSON: https://synthcore.org/api/idea/2504.json - API Markdown: https://synthcore.org/api/idea/2504.md - Verification status: mechanism_failed - Source: [arXiv:2608.18148](https://arxiv.org/abs/2608.18148) - Category: training - Solves: accuracy, stability, speedup - ML areas: ssm, rnn, attention, training-dynamics, scheduler - Math tags: pde, statistics, stochastic-processes, dynamical-systems, numerical-analysis - Ratings: usefulness 7/10; difficulty 4/10; novelty 7/10 ## Idea description Choose the neural operator's input-history length from the measured correlation time of the unresolved closure signal produced by coarse-graining. This avoids under-memory, which causes systematic closure error, and over-memory, which increases attention cost and can destabilize training. The same diagnostic can drive adaptive memory truncation across physical regimes. ## Mathematical statement Let $\mathcal C$ be a linear coarse-graining map, $\bar u=\mathcal C u$ the resolved field, and $\mathcal L$ the fine-scale PDE generator. Define the filtered closure-injection signal $r(t)=\mathcal C\mathcal L(u(t))-\mathcal L_{\mathrm{res}}(\bar u(t))$, where $\mathcal L_{\mathrm{res}}$ is the resolved-only drift obtained by inserting $\bar u$ into the coarse PDE. The unresolved contribution $r(t)$ is the forcing that a memory model must represent. Estimate its normalized autocorrelation $\rho(\tau)=\mathbb E[\langle r(t),r(t+\tau)\rangle]/\mathbb E[\|r(t)\|^2]$. Select $T_{\mathrm{in}}$ as the first lag at which $|\rho(\tau)|\leq\varepsilon$ and remains below the threshold over a persistence interval, or use the integrated correlation time $\tau_c=\int_0^\infty|\rho(\tau)|d\tau$ with safety factor $\alpha$. The mechanism predicts diminishing returns once the history exceeds the closure decorrelation time. ## Key formulas - $$r(t)=\mathcal C\mathcal L(u(t))-\mathcal L_{\mathrm{res}}(\bar u(t)),\qquad\bar u(t)=\mathcal C u(t).$$ - $$\rho(\tau)=\frac{\mathbb E_t[\langle r(t),r(t+\tau)\rangle_\Omega]}{\mathbb E_t[\|r(t)\|_\Omega^2]},\qquad\|r\|_\Omega^2=\int_\Omega\|r(x)\|^2dx.$$ - $$T_{\mathrm{in}}=\min\{\tau:|\rho(s)|\leq\varepsilon\text{ for all }s\in[\tau,\tau+\Delta]\}.$$ - $$\tau_c=\int_0^\infty|\rho(\tau)|d\tau,\qquad T_{\mathrm{in}}=\alpha\tau_c.$$ ## Implementation notes Use the selected window length to set the number of lag tokens in a causal attention operator, temporal convolution, recurrent state, or neural state-space model. For each high-resolution trajectory, compute the resolved field and the closure injection. Pseudocode is: `u_bar[t] = coarse(u_fine[t])`; `r[t] = coarse(fine_rhs(u_fine[t])) - resolved_rhs(u_bar[t])`; `rho[k] = mean(dot(r[t], r[t+k])) / mean(norm(r[t])**2)`; choose the first persistent threshold crossing; `m = ceil(T_in / dt)`; train the model using the previous $m$ slices only. The closure-residual definition and correlation-time criterion are obtained from the paper's coarse-graining mechanism; the threshold $\varepsilon$, persistence interval $\Delta$, and safety factor $\alpha$ must be estimated or cross-validated. If fine-scale right-hand sides are unavailable, estimate them by finite differences of high-resolution trajectories and fit the resolved drift separately. The first cheap experiment should use filtered Kuramoto–Sivashinsky or Burgers data and compare windows $m\in\{1,2,4,8,16,32\}$ with the diagnostic-predicted $m_*$. Measure one-step loss, 100-step rollout error, long-run spectra, and inference cost. The predicted signature is an elbow: error decreases while $T_{\mathrm{in}}