Constructive synthesis of neural networks realizing universal approximation results for PDE/PIDE solutions

Construct explicit procedures to synthesize neural networks that approximate solutions to high-dimensional nonlinear parabolic partial differential equations (PDEs) and partial integro-differential equations (PIDEs) in the sense guaranteed by existing universal approximation theorems, rather than merely asserting existence without a constructive method.

Background

The paper surveys results establishing that deep neural networks can approximate solutions of high-dimensional PDEs and PIDEs, often without suffering from the curse of dimensionality. These results demonstrate existence but are non-constructive, leaving practitioners without a clear algorithm to build the achieving network.

The authors highlight this gap and motivate their randomized deep splitting method as an approach with a full error analysis; however, the general issue of turning abstract existence guarantees into constructive network synthesis procedures remains an unresolved question in the broader literature.