Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do not provide enough information to derive tight deterministic and guaranteed bounds on function space norms. In this work, we move beyond a purely black-box setting and exploit the neural network structure directly. We present a framework for the certified and accurate computation of integral quantities of neural networks, including Lebesgue and Sobolev norms, by combining interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement and quadrature-based aggregation. On each box, we compute guaranteed lower and upper bounds for function values and derivatives, and propagate these local certificates to global lower and upper bounds for the target integrals. Our analysis provides a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, yielding certified computation of $L^p$, $W^{1,p}$, and $W^{2,p}$ norms. We further show how these ingredients lead to practical certified bounds for PINN interior residuals. Numerical experiments illustrate the accuracy and practical behavior of the proposed methods.

翻译：偏微分方程的神经网络方法需要在函数空间范数下具备可靠的误差控制。然而，训练后的神经网络通常只能在有限个点值处进行探测。若无强假设，仅凭点值评估无法提供足够信息来推导函数空间范数上紧致的确定性且有保证的界。在本工作中，我们超越了纯黑盒设定，直接利用神经网络结构。我们提出了一个框架，用于可验证且精确地计算神经网络的积分量（包括勒贝格和索伯列夫范数），其方法是将轴对齐箱体上的区间算术包络与自适应标记/细化以及基于数值积分的聚合相结合。在每个箱体上，我们计算函数值及导数的有保证下界和上界，并将这些局部证明传播到目标积分的全局下界和上界。我们的分析为此类可验证自适应数值积分过程提供了一个通用的收敛定理，并将其具体应用于函数值、雅可比矩阵和海森矩阵，从而实现了 $L^p$、$W^{1,p}$ 和 $W^{2,p}$ 范数的可验证计算。我们进一步展示了这些要素如何为PINN内部残差带来实用的可验证界。数值实验说明了所提方法的精度和实际表现。