Replace derivative-based latent-dynamics fitting with an integral regression and maintain a history stack selected by the smallest eigenvalue of its information matrix. The model should perform aggressive parameter updates only when the estimated latent regressors are sufficiently exciting, while a perturbation bound prevents false excitation caused by inaccurate hidden-state estimates.
Replace uniform minibatch sampling by a fixed-size determinantal point process whose similarity matrix is a diffusion kernel on the training-data k-NN graph. The sampler repels nearby or redundant examples while preserving multiple diffusion modes, so a small batch should cover intrinsic data geometry and provide lower-variance estimates of losses and gradients.
Use a spectral eigenvalue-counting function to bracket each target mode before neural optimization. The network then solves only within an interval containing exactly one eigenfrequency, preventing optimization from repeatedly collapsing to the lowest mode or jumping between modes.
Paper: A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic ResponsearXiv:2608.28683
Use the paper's finite-dimensional second-moment equations to compute the stationary covariance induced by a Markov-switched recurrent layer before training, then whiten or scale each mode's hidden state using that covariance. This can prevent mode-specific saturation and eliminate a long burn-in period in long-context RNNs and state-space models.