最小二乘残差最小化何时能求解微分方程？
When does least-squares residual minimization solve a differential equation?
- Universidad de Alicante（阿利坎特大学）
- The University of Texas at Austin（德克萨斯大学奥斯汀分校）
- UCLA（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究最小二乘残差最小化求解微分方程的条件，揭示驻点与极小化序列可能不收敛到真解，并给出恢复等价性的条件。
AI中文摘要：
最小二乘残差最小化通过一个优化问题来逼近微分方程。本文研究了从残差泛函的驻点和极小化序列过渡到精确解所需的条件。从残差与线性化算子值域之间的经典正交关系出发，我们研究了边界条件和参数化变分如何决定在驻点处关于残差所能推断的内容。即使对于适定问题，也可能出现光滑的伪临界点；对于线性方程，一个相容性条件给出了精确的投影刻画。对于具有跨声速边界数据的标量守恒律，每个极小化序列都收敛到一个不求解方程的连续函数，即使存在驻点熵激波。我们给出了逼近该极限迫使参数范数发散的条件。一个Hamilton-Jacobi例子表明，一个函数可以具有零残差而不是粘性解。对于粘性模型和一致凸Monge-Ampère临界点的有利结果，确定了恢复相应蕴含关系的条件。对于分段光滑试验族，定义残差泛函所需的正则性本身并不能证明训练中使用的采样梯度是合理的。移动的残差跳跃可能贡献该梯度中不存在的项；一个显式例子具有恒为零的采样梯度和非零的连续导数。总之，这些结果阐明了驻点和采样梯度对残差意味着什么，以及在什么条件下极小化序列会导致预期解。
英文摘要：
Least-squares residual minimization approximates a differential equation through an optimization problem. This paper investigates the conditions required to transition from stationary points and minimizing sequences of the residual functional to the exact solution. Starting from the classical orthogonality relation between the residual and the range of the linearized operator, we study how boundary conditions and parameterized variations determine what can be inferred about the residual at a stationary point. Smooth spurious critical points can occur even for well-posed problems; for linear equations, a compatibility condition yields an exact projection characterization. For a scalar conservation law with transonic boundary data, every minimizing sequence converges to a continuous function that does not solve the equation, even when a stationary entropy shock exists. We give conditions under which approaching this limit forces the parameter norms to diverge. A Hamilton--Jacobi example shows that a function can have zero residual without being the viscosity solution. Favorable results for a viscous model and uniformly convex Monge--Ampère critical points identify conditions that restore the corresponding implications. For piecewise-smooth trial families, the regularity needed to define the residual functional does not by itself justify the sampled gradient used in training. Moving residual jumps can contribute terms absent from that gradient; an explicit example has identically vanishing sampled gradients and a nonzero continuous derivative. Together, these results clarify what stationarity and sampled gradients imply about the residual, and under which conditions minimizing sequences lead to the intended solution.
