Global Basin Continuation for Neural Dynamics

1. Integration point: use a tanh GRU, vanilla RNN, or diagonal state-space model whose hidden state is iterated for a fixed input context until it reaches a recurrent regime. Select a task-relevant attractor by continuing the unperturbed trajectory. For chaotic or nonconvergent systems, replace point-attractor classification by clustering terminal windows or recurrence statistics. At each training checkpoint, perturb the hidden state with N samples from a prescribed ball or empirical hidden-state distribution.

2. Pseudocode: save the clean hidden state z_star; sample z_0^(k)=z_star+epsilon xi_k, where xi_k are normalized random vectors; iterate z_{t+1}=F_{theta,u}(z_t) for T steps; cluster terminal states or compare them with the clean attractor; estimate B_hat_0; add lambda times max(0,B_min-B_hat_0)^2 to the task loss, or reject the optimizer step if the basin fraction falls by more than delta. Repeat for a coarse grid of operating conditions u and perturbation amplitudes epsilon.

3. Computed versus estimated: the update map F_{theta,u} and task loss are computed exactly by the network. Attractors, basin labels, and B_i are estimated empirically from finite rollouts; use Wilson confidence intervals and increase N only near a suspected transition. Warm-start perturbation trajectories from neighboring parameter-grid points, while retaining all discovered attractor clusters instead of following only the nearest one.

4. First cheap experiment: train a two-layer tanh RNN on sequential MNIST or the adding-two-numbers task, comparing Adam, Adam plus spectral normalization, and Adam plus the basin penalty. At checkpoints, measure B_hat_0(epsilon) for epsilon from 10^-3 to 1, the number of hidden-state attractor clusters, long-horizon task error, and the local Jacobian spectral radius. The falsifiable prediction is that long-horizon failures begin when B_hat_0 drops sharply, often before the nominal Jacobian crosses spectral radius one. Across seeds, the perturbation threshold epsilon_c satisfying B_hat_0(epsilon_c)=0.5 should predict rollout failure within 20 percent. The basin penalty should shift epsilon_c upward and suppress abrupt attractor switching, even if the local Jacobian margin changes little.