标题： 线性自伴椭圆PDE在非线性逼近空间中的能量最小化收敛框架 显示英文标题

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

摘要： 近年来，出现了用于求解偏微分方程（PDEs）的非线性方法，例如物理信息神经网络（PINNs）。虽然这些方法在实践中通常表现良好，但其理论分析仍然有限，尤其是在收敛保证方面。本工作开发了一个通用的优化框架，用于解决来自线性自伴椭圆PDEs的能量最小化问题，该问题是在非线性但解析可处理的近似空间上提出的。该框架允许在线性和非线性参数之间进行自然划分，并支持混合优化策略：线性变量通过线性求解或最速下降进行更新，而非线性变量则通过约束投影下降进行处理。在对离散能量泛函的模块化结构假设下，包括可微性、有界性、正则性和方向凸性，我们建立了所得到算法的局部和全局收敛性。这些假设以抽象形式陈述，使得该框架可以应用于广泛的非线性近似流形。在一篇配套论文中，我们引入了一个基于重叠自由节点张量积B样条的具体空间，该空间满足所需的假设，并能够实现几何自适应求解器，具有严格的收敛保证。 显示英文摘要

摘要： 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, 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.