Residual neural networks overcome the curse of dimensionality for semilinear heat equations

arXiv:2609.03626v1 Announce Type: cross Abstract: Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist etain(0,infty) and ResNets Psi_{d,arepsilon}, dinN, arepsilonin(0,1], with at most eta d^{eta}arepsilon^{-eta} parameters whose realizations approximate the solution in dimension d with an L^2-error of at most arepsilon. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every xi>0, the explicit bound C_xi d^{4+xi}arepsilon^{-(3+xi)} on the number of parameters.

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Source: arXiv cs.LG | 2026-09-04