Adaptive reset neural ODE

Integrate this at the rollout module of a neural ODE, latent ODE, or continuous-time state-space model. Maintain a list of local vector fields \(f_{\Theta_k}(x,t)\), each responsible for an interval \([\tau_k,\tau_{k+1}]\). During training, initialize \(\tau_0=0\), choose a minimum window of \(s_{\min}\) discrete steps and a maximum cap \(S\), and warm-start \(\Theta_k\) from \(\Theta_{k-1}\). Optimize the teacher-forced window loss \(L_k=\sum_{b,s}\|\hat{x}_{b,k}(s)-x_b^{teacher}(\tau_k+s)\|_2^2\). Then roll the candidate model forward over a held-out batch. For every offset \(s\), compute \(e_b(s)=\|x_b^{teacher}(\tau_k+s)-\hat{x}_{b,k}(s)\|_2\), aggregate with the 95th or 99th percentile, and choose the first offset whose aggregate exceeds \(\varepsilon\); enforce \(s_{\min}\le s\le S\). Pseudocode: while \(\tau_k<T\), train \(\Theta_k\), evaluate \(\widehat E_k(s)\), set \(\tau_{k+1}\) using the stopping formula, append the module, and continue. At deployment, replace teacher resets by the model's predicted terminal state, which tests whether local accuracy prevents drift. The stopping rule and tolerance come from the paper; the batch quantile, discrete grid, and solver are empirical approximations. First experiment: compare one shared latent ODE with this adaptive ensemble on a chaotic oscillator or MuJoCo trajectory, using equal total parameters and solver evaluations. Measure long-horizon RMSE, number of windows, error immediately after every reset, and loss descent at equal FLOPs. Success means substantially slower error growth or a longer usable horizon at fixed compute.