Invariant-Guided Error-Compensating Rollouts

Integrate this at the rollout or inference loop of a neural world model, Hamiltonian neural network, or learned simulator rather than inside the network's parameter optimizer. The model must expose either two composable subflows $\Phi^{(1)}$ and $\Phi^{(2)}$, as in the paper's split evolution, or two vector-field components that can be advanced sequentially. Build a lightweight policy $\pi_\theta$ that receives $o_n=[t_n,\delta t_{n-1},(C(z_n)-C(z_0))/s_C,\operatorname{Var}(C(z_{n-k:n}))/s_C^2,\|z_n-z_{n-1}\|]$ and outputs a bounded action: $\delta t_n=\delta t_{\min}+(\delta t_{\max}-\delta t_{\min})\operatorname{sigmoid}(a_n)$. Pseudocode: initialize $z=z_0$, $t=0$; for $n=0,...,N-1$, compute $a=\pi_\theta(o)$, set $\delta t$ from the bounded action, apply subflow 1 for $\delta t/2$, subflow 2 for $\delta t$, then subflow 1 for $\delta t/2$, compute $C(z)$, assign $r_n=-|C(z)-C(z_0)|/s_C-\lambda\delta t_{\min}/\delta t$, update the policy with PPO or a critic, and continue until the time horizon. The mathematical quantities are the symmetric split composition, invariant-drift reward, and terminal long-horizon penalty; estimate $s_C$ from the training rollout distribution and estimate the variance feature using a rolling window. Train on small systems or short trajectories where exact dense rollouts are available, but do not give the policy the exact target state. First test on a 2D harmonic oscillator or spring-mass world model with a fixed compute budget, comparing fixed-step leapfrog or RK4, standard local adaptive stepping, and the learned controller. Measure invariant drift versus number of function evaluations, long-horizon state error, and rollout stability. Success is a lower energy-drift curve at equal evaluations or at least 25% fewer evaluations at equal terminal error; additionally test transfer to unseen initial conditions and to a larger system.