numerical analysis

Neural Very Weak Formulations enabling Hardware-Oriented deep PDE solvers

arXiv:2607.14498

summary

The paper demonstrates that least‑squares very weak formulations of elliptic PDEs can be discretized with low‑regularity neural networks, using simple step or one‑bit quantized activations that are suitable for efficient hardware implementation.

Abstract

We show, as a proof of concept, that least-squares very weak formulations of elliptic problems can be effectively discretized by neural networks possessing low regularity, provided the test functions are drawn from appropriately smooth spaces. Apart from the immediate computational benefit of avoiding automatic differentiation, this approach, evaluated across various neural network spaces, demonstrates good performance even in challenging contexts, such as singular solutions and high dimensional settings. Particular attention is paid to trial functions based on step activations and one bit quantized linear functions, which are amenable to efficient hardware-oriented implementations.

Topics & keywords

#deep learning for pdes#very weak formulation#least-squares discretization#hardware‑efficient neural nets#quantized activations#high‑dimensional elliptic problemsvery weak formulationleast-squareselliptic PDEstep activationbinary quantizationhardware acceleration
Neural Very Weak Formulations enabling Hardware-Oriented deep PDE solvers · wovepaper