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

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 η∈(0,∞) and ResNets Ψd,ε, d∈N, ε∈(0,1], with at most ηdηε−η parameters whose realizations approximate the solution in dimension d with an L2-error of at most ε. 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 ξ>0, the explicit bound Cξd4+ξε−(3+ξ) on the number of parameters.