Construct a periodically driven hybrid recurrent state-space model whose vector field is piecewise smooth across learned switching surfaces. Engineer a transverse homoclinic intersection around a hyperbolic recurrent state; the resulting shift-like invariant set provides a controllable symbolic reservoir for sequence prediction and long-horizon generation.
Replace step-by-step hidden-state storage in a latent ODE, state-space model, or world model with a polynomial trajectory represented independently on short time blocks. At the end of each block, pass the next hidden state by summing temporal coefficients, allowing training and inference to discard the completed block while retaining a mathematically exact block interface.
Replace a dense token or channel mixing matrix by a fixed sparse directed graph whose states are ordered pairs of symbols and whose transitions advance through a cyclic phase. Each state has exactly two allowed successors, obtained by appending a symbol different from the previous two, producing a strongly connected, vertex-transitive sparse mixer with shared local dynamics. The prescribed phase structure prevents arbitrary short-cycle routing and can act as an anti-collapse inductive bias in…
Insert a rearrangement operation on scalar feature maps sampled along an ordered coordinate such as time, spatial position, or a neural-field input grid. The operation sorts values into non-increasing order, preserving the empirical histogram exactly while provably not increasing the Riesz fractional variation in the ideal one-dimensional continuous setting.
Prepend an adaptive Savitzky-Golay derivative bank to a temporal neural network. For each input channel and derivative order, select the local window by minimizing Stein's unbiased risk estimate, then concatenate the raw signal with the estimated derivatives. This supplies denoised velocity and acceleration features without requiring clean derivative targets or forcing the backbone to learn unstable finite-difference filters.
Generate temporal attention or convolution weights with the Graham–Knuth–Patashnik recurrence instead of learning every lag weight independently. For nonnegative recurrence parameters, the resulting lag sequence is strongly log-concave, so its normalized kernel is naturally unimodal and suppresses high-frequency sign-free oscillations without requiring a separate smoothness penalty. The six parameters can be learned per head, channel group, or layer, giving O(1) learned parameters for an…
Use the complex-conjugate palindromic coefficient that cancels the leading temporal phase defect of oscillatory modes. Implement complex arithmetic directly or use an exactly equivalent doubled-real state, then project the final state to its real component for real-valued prediction tasks.
Add a fixed or weakly learned interface-localized branch to a sequence model. Set the critical mass term to zero and make the transport coefficient change sign across a learnable interface, producing a localized mode that pools information near a detected transition rather than averaging uniformly over the sequence.
Insert a fixed or lightly gated lower-triangular binomial-transform layer into a sequence model to create global interactions across positions without forming attention logits. For a sequence of length N, mix each output position with all earlier positions using coefficients determined by an integer m; initialize the layer fixed and optionally learn a diagonal channel gate or a small mixture over m values.
Paper: $m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distributionarXiv:2608.12011