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

What the math gives to ML

The paper gives a constructive residual-network realization of a multilevel Picard estimator for semilinear heat equations, rather than only proving existence of an approximating network. Its transferable asset is an explicit state-carrying computation graph: shortcut channels preserve the spatial/context representation and an accumulator, while successive residual branches add estimator corrections. This suggests solver-like neural modules for high-dimensional regression, iterative latent prediction, and diffusion-related computation, where additive corrections may be more stable and parameter-efficient than repeatedly rebuilding the full state. The theorem also supplies a falsifiable hypothesis that parameter growth can remain polynomial in input dimension and inverse accuracy.