We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^2$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.

翻译：我们针对线性偏微分方程的物理信息神经网络（PINNs）证明了先验和后验误差估计。分析了原形式与混合形式的椭圆方程、弹性力学方程、抛物型方程、双曲型方程、斯托克斯方程以及一个偏微分方程约束优化问题。在分析过程中，我们提出了一个基于双线性形式共同语言的抽象框架，并证明了强制性与连续性可导出误差估计。所得估计具有尖锐性，揭示了PINN公式中针对初始条件和边界条件的$L^2$罚函数方法会削弱误差衰减范数。最后，利用PINN优化的最新进展，我们给出了展示该方法实现精确解能力的数值算例。