Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods, however, is ultimately constrained by the limitations of existing computation models. Since digital computers constitute the primary physical realizations of numerical computation, and Turing machines define their theoretical limits, the question of Turing computability of PDE solutions arises as a problem of fundamental theoretical significance. The Turing computability of PDE solutions provides a rigorous framework to distinguish equations that are, in principle, algorithmically solvable from those that inherently encode undecidable or non-computable behavior. Once computability is established, complexity theory extends the analysis by quantifying the computational resources required to approximate the corresponding PDE solutions. In this work, we present a novel framework based on least-squares variational formulations and their associated gradient flows to study the computability and complexity of PDE solutions from an optimization perspective. Our approach enables the approximation of PDE solution operators via discrete gradient flows, linking structural properties of the PDE, such as coercivity, ellipticity, and convexity, to the inherent complexity of their solutions. This framework characterizes both regimes where PDEs admit effective numerical solvers in polynomial-time and those exhibiting complexity blowup, where the input data possess polynomial-time complexity, yet the solution itself scales super-polynomially.

翻译：偏微分方程（PDEs）是描述物理现象的基本工具，然而大多数具有实际意义的偏微分方程无法解析求解，需要借助数值近似方法。然而，此类数值方法的可行性最终受限于现有计算模型的局限性。由于数字计算机是数值计算的主要物理实现方式，而图灵机定义了其理论极限，因此偏微分方程解的图灵可计算性问题成为一个具有基础理论意义的研究课题。偏微分方程解的图灵可计算性提供了一个严谨的框架，用以区分在原则上可通过算法求解的方程与那些本质上编码了不可判定或不可计算行为的方程。一旦可计算性得以确立，复杂性理论通过量化近似求解相应偏微分方程解所需的计算资源，进一步扩展了分析维度。在本研究中，我们提出了一种基于最小二乘变分形式及其伴随梯度流的新框架，从优化视角研究偏微分方程解的可计算性与复杂性。我们的方法通过离散梯度流实现对偏微分方程解算子的近似，将偏微分方程的结构性质（如强制性、椭圆性和凸性）与其解的固有复杂性联系起来。该框架既刻画了偏微分方程存在多项式时间有效数值求解器的情形，也描述了复杂性爆炸的机制——即输入数据具有多项式时间复杂性，而解本身的规模却呈现超多项式增长。