Dimension-free shallow network approximation via spectral barron regularity for the high-dimensional biharmonic equation

Source abstract

Membership of a function in a Barron space is the mechanism that allows a single–hidden–layer neural network to approximate it at the dimension-free Monte Carlo rate n − 1 ∕ 2 , thereby escaping the curse of dimensionality that afflicts grid- and mesh-based methods, whose cost grows like ε − d ∕ s . Existing Barron-regularity results for partial differential equations are confined to second-order elliptic operators. We treat the fourth-order biharmonic operator Δ 2 on the cube [ 0 , 1 ] d with simply supported boundary conditions, for which the operator is a diagonal Fourier multiplier in the sine basis. We prove that if the load f lies in the spectral Barron space F s then the solution u lies in F s + 4 , with the sharp bound ‖ u ‖ F s + 4 ≤ π − 4 ( 1 + d − 1 ∕ 2 ) 4 ‖ f ‖ F s . Strikingly, this constant is not merely polynomial in d : it is uniformly bounded and monotonically decreasing in d , converging to π − 4 . Consequently, the solution is approximated to accuracy ε in the H 2 energy norm by a shallow cosine network with n = O ( ε − 2 ) neurons, with a constant independent of d . A numerical study on canonical Barron targets in dimensions d ∈ { 5 , 10 , 20 , 40 } confirms the predicted n − 1 ∕ 2 rate and its flatness in d . The result is a model problem for the fourth-order operators of thin-structure mechanics (Kirchhoff plates) and phase-field models (Cahn–Hilliard).