Certified and accurate computation of function space norms of deep neural networks

2026-03-06

AI Summary

This work addresses the lack of reliable, deterministic error control in existing neural network approaches for solving partial differential equations with respect to function space norms. The authors propose a novel framework that integrates interval arithmetic, adaptive domain partitioning, quadrature-based numerical integration, and automatic differentiation with Hessian propagation. For the first time, this framework enables certified, adaptive, and convergent computation of global upper and lower bounds for deep neural networks in Lebesgue (L^p) and Sobolev (W^{1,p}, W^{2,p}) norms. By moving beyond conventional black-box evaluation paradigms, the method provides rigorous error control over the internal residuals of physics-informed neural networks (PINNs). Numerical experiments demonstrate the high accuracy and reliability of the proposed approach.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Adversarial Attacks & Robustness
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

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.

Problem

• function space norms
• certified computation
• neural networks
• error control
• PDEs

Innovation

• certified computation
• function space norms
• interval arithmetic
• adaptive quadrature
• physics-informed neural networks