We provide an a priori analysis of a certain class of numerical methods, commonly referred to as collocation methods, for solving elliptic boundary value problems. They begin with information in the form of point values of the right side f of such equations and point values of the boundary function g and utilize only this information to numerically approximate the solution u of the Partial Differential Equation (PDE). For such a method to provide an approximation to u with guaranteed error bounds, additional assumptions on f and g, called model class assumptions, are needed. We determine the best error (in the energy norm) of approximating u, in terms of the number of point samples m, under all Besov class model assumptions for the right hand side $f$ and boundary g. We then turn to the study of numerical procedures and asks whether a proposed numerical procedure (nearly) achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing a specified data driven loss function over a set $\Sigma$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a correct data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a loss function $L^*$ is proposed. We also address the recent methods of Physics Informed Neural Networks (PINNs). Minimization of the new loss $L^*$ over neural network spaces $\Sigma$ is referred to as consistent PINNs (CPINNs). We prove that CPINNs provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $\Sigma$ has sufficient approximation capabilities. Finally, numerical examples illustrating the benefits of the CPINNs are given.

翻译：本文对一类常称为配点法的数值方法进行了先验分析，用于求解椭圆型边值问题。该方法以方程右端项f的离散点值及边界函数g的离散点值作为输入信息，仅依赖这些信息对偏微分方程（PDE）的解u进行数值逼近。要使此类方法在保证误差界的前提下提供u的近似解，需对f和g施加额外假设（称为模型类假设）。我们确定了在右端项f和边界g的所有Besov类模型假设下，基于采样点数m逼近u的最佳误差（按能量范数度量）。继而转向数值方法研究，探讨所提出的数值方法能否（近似）达到最优恢复误差。我们分析了一类数值方法：通过在集合Σ（该集合可为有限维线性空间或更一般的有限维流形）上最小化特定数据驱动损失函数，生成u的数值逼近。研究表明，此类方法的成功关键取决于能否选择与PDE相容且具备精确误差控制能力的数据驱动损失函数。基于此分析，我们提出了一种损失函数L^*。本文还探讨了近期兴起的物理信息神经网络（PINNs）方法。将新损失L^*在神经网络空间Σ上最小化的方法称为相容性PINNs（CPINNs）。我们证明：若优化问题可数值实现且Σ具有充分逼近能力，则CPINNs能实现解u的最优恢复。最后给出数值算例验证CPINNs的优势。