Turn an iterative optimization or equilibrium computation inside a neural network into a differentiable layer whose backward pass solves the implicit adjoint system with conjugate gradients or GMRES using only automatic-differentiation matrix-vector products. This avoids storing unrolled iterations and avoids explicit Hessian or Jacobian construction, enabling longer solver horizons and lower-memory implicit architectures.
Replace uniform collocation for a fixed random-feature neural PDE solver with sampling from the leverage-score density of the operator-applied features. Whiten the retained residual feature space before solving for output coefficients, so the sampled least-squares matrix has an identity-like expected Gram rather than inheriting severe anisotropy from the differential operator. The same construction can be used for a linearized neural network by treating Jacobian features as the trial functions.
For bandlimited or progressively higher-frequency targets, construct a nested random Fourier layer whose frequency window grows according to the paper's bandwidth laws instead of sampling all frequencies from one fixed range. Train with a low-bandwidth subset first, then activate additional frequency blocks. This creates a mathematically motivated spectral curriculum and may reduce early optimization difficulty while preserving high-frequency accuracy.
Use the paper's order-parameter dynamics to initialize spectral feature modes with deliberately separated activation times. This creates a controlled progressive-learning curriculum in which dominant modes become available first and weaker modes activate later, potentially reducing early gradient interference.
Initialize an invertible neural linear layer from a bounded discrete random matrix only after checking that every leading principal submatrix is nonsingular and that its LU growth factor is below a prescribed threshold. This replaces blind random initialization with a cheap resampling rule designed to prevent zero pivots and excessive finite-precision amplification in reversible or flow-based networks.