Training

Grids. Every input has its own grid (a buffer). Keep them on the data:

model.update_grid(x)          # pykan's adaptive grid: quantiles + a uniform share
model.refine_grid(10)         # finer grid, same function (exact for nested B-spline grids)
model[0].change_basis(kf.Chebyshev(12))   # any basis to any other, by least squares

All three refit the coefficients so the function is preserved; parameters are replaced, so create the optimiser again afterwards.

Regularisation (pykan): loss + lam * regularization(model, x) — L1 and entropy of edge magnitudes, optionally L1 and smoothness of coefficients; without x it uses |coef| (efficient-kan). Needed before pruning.

Initialisation: coef_init="lecun" / "glorot" give unit output variance for N(0, 1) inputs, accounting for the basis and the SiLU branch.

From an MLP: from_mlp(nn.Sequential(Linear, act, Linear, ...)) and KANLinear.from_linear(linear, act) reproduce the MLP exactly; the splines start at zero and learn corrections.

Optimisers: everything works with Adam; LBFGS (pykan's default for small problems) converges much faster on smooth regression and PDE tasks.