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：针对偏微分方程的神经网络方法需要在函数空间范数下具有可靠误差控制。然而，训练后的神经网络通常只能通过有限点值进行探测。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的紧致确定性保证界。本研究突破纯黑箱设定，直接利用神经网络结构特性，提出一个用于神经网络积分量可证精确计算的框架——通过结合轴对齐盒子上的区间算术包络、自适应标记/细化以及基于求积的聚合方法，实现勒贝格范数与索伯列夫范数的可证精确计算。在每个子盒上，我们计算函数值和导数的保界上下界，并将这些局部证书传播为目标积分的全局上下界。我们的分析为该类可证自适应求积流程提供了通用收敛定理，并将其具体应用于函数值、雅可比矩阵和海森矩阵，实现$L^p$、$W^{1,p}$和$W^{2,p}$范数的可证计算。进一步展示了这些要素如何为物理信息神经网络内部残差提供实用的保界界。数值实验验证了所提方法的精确性与实际表现。