Life sciences · Preprint

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

arXiv · September 3, 2026

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AI Summary

This is a preprint in computational mathematics proving theoretical bounds on the parameter complexity of residual neural networks for solving high-dimensional semilinear heat equations. It is not an empirical study, clinical trial, or evidence relevant to health care practice, and has not been peer reviewed.

What was studied

Preprint.

Key findings

ResNets can overcome curse of dimensionality for semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities under stated conditions Parameter bound: at most ηd^η ε^{-η} parameters achieve L^2-error of at most ε in dimension d For ridge-sum initial conditions and specific activations, explicit bound is C_ξ d^{4+ξ} ε^{-(3+ξ)} parameters for every ξ>0

Safety

Safety was not reported in the material analysed. Check the source before drawing any conclusion about harm.

Who this may help

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Plain-language summary

This is a theoretical mathematics paper proving computational properties of neural networks for PDEs, not an empirical study or clinical evidence, and does not address any health or medical question.

Reported figures

Quoted from the source exactly as published.

Parameter bound (general case)ηd^η ε^{-η}

Parameter bound (ridge-sum case)C_ξ d^{4+ξ} ε^{-(3+ξ)}

Approximation error (L^2)at most ε

Caveats

This is a preprint posted on arXiv and has not undergone peer review.
This is pure mathematics and computational theory; no empirical validation or application to clinical, biological, or medical problems is reported.
No numerical experiments or computational comparisons with other methods are described in the abstract.
The work addresses only semilinear heat equations with specific mathematical properties; generalizability to other PDEs or practical applications is not discussed.

Evidence assessment

Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.

Study design

insufficient

Not applicable; this is a mathematical theorem and proof, not an empirical study.

Directness

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Not applicable; no clinical or biological outcome is measured or targeted.

Consistency

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Not applicable; single theoretical contribution without empirical validation reported.

Risk of bias

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Sample adequacy

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Source 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 $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. 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 $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.

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Diyora SalimovaIlkhom Mukhammadiev