Non-Uniqueness of Solutions in Neural Variational Methods

TL;DR

This paper demonstrates that various neural variational methods can be ill-posed due to finite measurements, leading to non-unique solutions regardless of the continuous problem's well-posedness.

Contribution

It develops an analytical framework revealing why neural variational discretizations often result in ill-posed problems with non-unique solutions.

Findings

Finite measurements cause ill-posedness in neural variational methods.

Non-uniqueness of solutions occurs independently of the continuous problem's well-posedness.

The framework applies to multiple neural discretization approaches.

Abstract

Recent work has shown that strong-form physics-informed neural networks (PINNs) based on pointwise enforcement of differential operators can be ill-posed due to the combination of sufficiently expressive neural network trial spaces with finitely many measurements. In this work, we develop an abstract analytical framework that isolates this finite-information mechanism and extends its applicability beyond strong-form formulations. We apply the framework to three representative variational neural discretizations: the Deep Ritz method, neural network discretizations of variational regularization functionals, and weak PINNs. Despite their differing formulations, these methods constrain the neural trial function only through finitely many linear measurements, such as quadrature evaluations or finite-dimensional test spaces. We show that this structural feature leads to ill-posed discrete optimization problems, manifested by non-uniqueness or degeneracy of minimizers, independently of the well-posedness of the underlying continuous variational problem.