We propose a new loss function for supervised and physics-informed training of neural networks and operators that incorporates a posteriori error estimate. More specifically, during the training stage, the neural network learns additional physical fields that lead to rigorous error majorants after a computationally cheap postprocessing stage. Theoretical results are based upon the theory of functional a posteriori error estimates, which allows for the systematic construction of such loss functions for a diverse class of practically relevant partial differential equations. From the numerical side, we demonstrate on a series of elliptic problems that for a variety of architectures and approaches (physics-informed neural networks, physics-informed neural operators, neural operators, and classical architectures in the regression and physics-informed settings), we can reach better or comparable accuracy and in addition to that cheaply recover high-quality upper bounds on the error after training.

翻译：我们提出了一种新的损失函数，用于有监督训练和物理信息训练的神经网络与算子，该损失函数融入了后验误差估计。具体而言，在训练阶段，神经网络学习了额外的物理场，这些物理场在经过计算代价低廉的后处理阶段后，能够导出严格的误差上界。理论结果建立在函数后验误差估计理论之上，该理论允许对一类实际相关的偏微分方程系统地构建此类损失函数。从数值计算方面来看，我们在一系列椭圆型问题上证明，对于多种架构和方法（物理信息神经网络、物理信息神经算子、神经算子，以及回归和物理信息设置中的经典架构），我们可以达到更好或相当的计算精度，此外，在训练后还能以较低成本恢复高质量的误差上界。