Deep neural networks provably overcome curse of dimensionality for PDEs

Researchers have demonstrated that deep neural networks (DNNs) can overcome the curse of dimensionality when approximating solutions to Kolmogorov partial differential equations. This mathematical proof extends previous findings by showing that networks using ReLU, leaky ReLU, and softplus activation functions can achieve approximation accuracy without a prohibitive increase in computational cost relative to the problem's dimension. The work establishes this capability in the $L^p$-sense for a broad range of $p$ values. AI

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IMPACT Provides theoretical grounding for using deep learning to solve high-dimensional scientific computing problems.

RANK_REASON Academic paper presenting a theoretical proof for deep neural networks overcoming a computational challenge.

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• ReLU
• leaky ReLU
• softplus
• Kolmogorov partial differential equations
• curse of dimensionality
• deep neural networks
• arXiv

• paper
• other