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

2025-09-03 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning, Engineering & Applied Sciences · Depth: Expert, extended

Summary

This research introduces a framework for the certified and accurate computation of function space norms, specifically Lebesgue and Sobolev norms, for deep neural networks. Traditional methods for error control in neural network-based PDE solvers often rely on pointwise evaluations, which are insufficient for guaranteed bounds. The proposed framework moves beyond a black-box approach by directly exploiting the neural network's structure. It combines interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement strategies and quadrature-based aggregation. This process computes guaranteed lower and upper bounds for function values and derivatives, propagating these local certificates to global bounds for target integrals. The analysis includes a general convergence theorem for such certified adaptive quadrature procedures, instantiated for function values, Jacobians, and Hessians, yielding certified computation of $L^{p}$, $W^{1,p}$, and $W^{2,p}$ norms. The framework also provides practical certified bounds for PINN interior residuals, with numerical experiments demonstrating accuracy.

Key takeaway

For AI Scientists and Research Scientists developing or deploying neural network-based PDE solvers, this framework offers a critical advancement in error control. You can now obtain deterministic, guaranteed bounds on function space norms, moving beyond probabilistic error assessments. This enables more reliable validation of model performance and enhances trust in neural network solutions for sensitive applications, particularly in physics-informed neural networks (PINNs). Consider integrating this certified adaptive quadrature approach to ensure the robustness and accuracy of your neural network's integral quantities and Sobolev norms.

Key insights

Certified function space norms for neural networks are achieved via interval arithmetic and adaptive quadrature.

Principles

• Pointwise evaluations alone are insufficient for certified function space norm bounds.
• Interval arithmetic provides rigorous local bounds for neural network outputs and derivatives.
• Adaptive refinement strategies improve the convergence of integral error bounds.

Method

The method combines local interval arithmetic enclosures for neural network outputs and derivatives on axis-aligned boxes, adaptive refinement of boxes with high uncertainty, and quadrature rules to aggregate local certificates into global bounds for integrals and norms.

In practice

• Compute certified $L^{p}$, $W^{1,p}$, and $W^{2,p}$ norms for neural networks.
• Derive practical certified bounds for PINN interior residuals.
• Utilize activation pattern checks for exact integration on ReLU affine linear pieces.

Topics

• Deep Neural Networks
• Function Space Norms
• Certified Integration
• Interval Arithmetic
• Adaptive Quadrature

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.