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.

References

However, these results are abstract and only prove the existence of a neural network which can well approximate the given solution of the PDE or PIDE, however it is left open how to construct it.

— Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs  (2405.05192 - Neufeld et al., 2024) in Section 1 (Introduction)

Extensions to state-dependent coefficients, gradient-dependent nonlinearities, and space--time approximations, as well as sharper exponents and constructive training procedures, remain open.

— Residual neural networks overcome the curse of dimensionality for semilinear heat equations  (2609.03626 - Mukhammadiev et al., 3 Sep 2026) in Section Conclusion