We prove a priori and a posteriori error estimates, also known as the generalization error in the machine learning community, 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. Our results give insight into the potential of neural networks for high dimensional PDEs and into the benefit of encoding constraints directly in the ansatz class. The provided estimates are -- apart from the Poisson equation -- the first results of best-approximation and a posteriori error-control type. 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（在机器学习领域也称为泛化误差）。我们分析了原始形式和混合形式的椭圆方程、弹性方程、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。为进行分析，我们提出了一个基于双线性形式共同语言的抽象框架，并证明了强制性和连续性可导出误差估计。我们的研究结果揭示了神经网络在高维偏微分方程中的潜力，以及直接在ansatz类中编码约束条件的优势。除泊松方程外，所提供的估计是首个最佳逼近误差和后验误差控制类型的结果。最后，利用PINN优化的最新进展，我们给出了数值算例，展示了该方法获得精确解的能力。