Certified and accurate computation of function space norms of deep neural networks
| Authors | Johannes Gründler et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2603.06431 |
| Download | |
| Categories | cs.LG, stat.ML |
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 , 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.
Engineering Breakdown
Plain English
This paper solves a critical problem in using neural networks to solve PDEs: verifying that trained networks actually satisfy error bounds in continuous function spaces, not just at scattered test points. The authors developed a framework that combines interval arithmetic on axis-aligned boxes with adaptive refinement and quadrature to compute guaranteed upper and lower bounds on integral quantities like Lebesgue and Sobolev norms. Instead of treating the neural network as a black box and hoping point evaluations suffice, they exploit the network's internal structure to get certified, mathematically guaranteed error estimates. This bridges the gap between what we can actually measure (discrete point values) and what we need to prove (continuous function space behavior).
Core Technical Contribution
The core novelty is a structured approach to certified computation of function space norms for neural networks without requiring exhaustive sampling or strong smoothness assumptions. Rather than relying on sparse point evaluations or Lipschitz bounds alone, the method decomposes the domain into axis-aligned boxes and uses interval arithmetic to track guaranteed bounds on network outputs within each box. The framework combines three key components: (1) interval enclosures that rigorously bound neural network behavior on boxes, (2) adaptive marking and refinement to focus computation where uncertainty is highest, and (3) aggregation via quadrature rules to convert pointwise bounds into certified integral quantities. This is fundamentally different from prior approaches that either ignore the network architecture or assume global Lipschitz constants, which leads to loose bounds in practice.
How It Works
The method operates in phases. First, the input domain is partitioned into axis-aligned boxes, and for each box the neural network's output range is enclosed using interval arithmetic—computing lower and upper bounds for all possible outputs over that box given the network weights and biases. This enclosure respects the network's computation graph structure, propagating interval bounds through each layer. Second, an adaptive refinement loop marks boxes with large uncertainty (gap between upper and lower bounds) and subdivides them, concentrating computational effort where it matters most. Third, once boxes are sufficiently refined, the method uses numerical quadrature rules (like Gaussian quadrature) to aggregate the interval-bounded values into certified bounds on integral quantities: the user can compute a lower bound by integrating lower bounds and an upper bound by integrating upper bounds. The output is a deterministic certificate: 'this Sobolev norm is guaranteed to lie in [L, U]' with mathematical rigor, not just probabilistic guarantees.
Production Impact
For engineers deploying neural networks as PDE solvers (common in physics-informed neural networks, scientific computing), this provides missing verification machinery. In production, you'd integrate this as a post-training certification step: after training your network, run this framework to generate a certificate proving 'this network's L2 error is at most ε in the domain of interest.' This solves real compliance and safety issues in high-consequence domains like structural analysis or climate modeling where you cannot ship an uncertified solver. The trade-off is computational cost: adaptive refinement requires many interval enclosure evaluations (one per box per layer) and quadrature evaluations, so this is a post-hoc verification tool, not a cheap add-on—expect 10-100× slowdown versus raw inference. Integration complexity is moderate: you need to implement interval arithmetic for your network architecture and handle adaptive refinement bookkeeping, but the core mathematics is straightforward. The requirement for axis-aligned boxes also means the method scales better in low to moderate dimensions (typical for PDE meshes) than in high-dimensional spaces.
Limitations and When Not to Use This
The method assumes the domain can be partitioned into axis-aligned boxes, which becomes impractical in very high dimensions (curse of dimensionality still applies) or on irregular geometric domains without significant preprocessing. It does not handle networks with architecture choices that complicate interval propagation (e.g., division, trigonometric functions, or complex branching logic), limiting applicability to feedforward networks with standard activations. The paper assumes you have access to the trained network structure and weights; it cannot certify black-box neural networks where you only observe outputs. The adaptive refinement process is greedy and may still require substantial computation for tight bounds; pathological networks with highly non-monotonic behavior on boxes could require exponentially many refinements. The framework also implicitly assumes the network is deterministic and differentiable in the interval sense; it does not directly address stochastic networks or those with discrete components.
Research Context
This work builds on decades of research in interval arithmetic (dating to Moore, 1960s) and certified computing, applying it to the modern problem of neural network verification. It extends prior work on verified bounds for neural networks (e.g., Zonotope-based methods, abstract interpretation) by focusing specifically on integral quantities and function space norms rather than pointwise values. The paper is part of a growing ecosystem around 'physics-informed neural networks' (PINNs) and their verification, complementing work on PINN training and sampling strategies. It opens the research direction of certified surrogate models in scientific computing: can we build and certify neural network surrogates that practitioners can trust as replacements for traditional numerical solvers?
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