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 $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η...
| Authors | Ilkhom Mukhammadiev, Diyora Salimova |
|---|---|
| Primary category | math.NA |
| Published | 3 September 2026 |
| Last updated at source | 3 September 2026 |
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