Residual-aware adaptive training step

(1) Integration point: apply during inference and training rollouts of a neural ODE, learned PDE solver, world model, or latent state-space model, immediately before each Euler or Runge-Kutta step. Evaluate the velocity tensor $v=f_\theta(x,t)$ and select the next step from its maximum norm. (2) Pseudocode: input state $x$, grid spacing $h_{\min}$, network $f_\theta$; evaluate $v=f_\theta(x,t)$; calculate $s=\max_j\|v_j\|_2$; set $dt=\min(dt_{\max},c_{CFL}h_{\min}/(s+\delta))$; clamp to $dt_{\min}$; advance with Euler, $x\leftarrow x+dt\,v$, or use the same $dt$ in all stages of an RK step; repeat until the target time. Initially detach $s$ and $dt$ from autograd to avoid unstable gradients through the controller. (3) The CFL form and safety factor come from the paper; $h_{\min}$, $dt_{\min}$, $dt_{\max}$, and $\delta$ are engineering choices. Log accepted step sizes and the nondimensional displacement ratio $r=dt\,s/h_{\min}$, which should remain below approximately $c_{CFL}$. (4) First experiment: compare fixed-step Euler and adaptive-step Euler on a small learned 2-D advection or reaction-diffusion model, using identical networks and target rollout time. Evaluate steps per rollout, wall-clock inference, trajectory error against a high-resolution reference, and instability frequency. A useful success criterion is at least 2x fewer steps on smooth trajectories at comparable error, with no additional blow-ups on high-velocity cases.