Core Concepts
Deep neural networks can overcome the curse of dimensionality in approximating solutions of certain nonlinear PDEs.
Abstract
The study proves that deep learning algorithms can tackle high-dimensional PDEs effectively. It highlights theoretical results supporting the effectiveness of deep learning in overcoming the curse of dimensionality. The main contribution is proving that deep neural networks can handle nonlinear PDEs with gradient-dependent nonlinearities. The paper introduces a mathematical framework for DNNs and provides rigorous proofs for their capabilities in approximating solutions without suffering from dimensionality issues.
Stats
The number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension and the reciprocal of the prescribed accuracy.
Deep neural networks can approximate initial conditions, linear parts, and nonlinear parts without curse of dimensionality.
Realizations of MLP approximations can be represented by DNNs.