A convergence framework for energy minimisation of linear self-adjoint elliptic PDEs in nonlinear approximation spaces

TL;DR

This paper introduces a flexible convergence framework for energy minimisation in linear self-adjoint elliptic PDEs using nonlinear approximation spaces, providing theoretical guarantees for hybrid optimisation strategies.

Contribution

It develops a general abstract framework for convergence analysis of energy minimisation algorithms in nonlinear approximation spaces for elliptic PDEs, accommodating hybrid optimisation methods.

Findings

Established local and global convergence under structural assumptions.

Framework applies to a broad class of nonlinear approximation manifolds.

Supports hybrid optimisation strategies with theoretical guarantees.

Abstract

Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis remains limited, especially regarding convergence guarantees. This work develops a general optimisation framework for energy minimisation problems arising from linear self-adjoint elliptic PDEs, formulated over nonlinear but analytically tractable approximation spaces. The framework accommodates a natural split between linear and nonlinear parameters and supports hybrid optimisation strategies: linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using constrained projected descent. We establish both local and global convergence of the resulting algorithm under modular structural assumptions on the discrete energy functional, including differentiability, boundedness, regularity, and directional convexity. These assumptions are stated in an abstract form, allowing the framework to apply to a broad class of nonlinear approximation manifolds. In a companion paper [Magueresse, Badia (2025, arXiv:2508.17705)], we introduce a concrete instance of such a space based on overlapping free-knot tensor-product B-splines, which satisfies the required assumptions and enables geometrically adaptive solvers with rigorous convergence guarantees.