Function-Space Certification for Neural Operators
Abstract
Certifying the robustness and physical consistency of scientific machine-learning models ensures that their predictions remain reliable beyond the test set. For neural operators, which learn PDE solution operators by mapping infinite-dimensional function spaces, certification means establishing rigorous, domain-wide guarantees for their output functions. However, standard neural network (NN) verification methods are fundamentally designed for vector-to-vector mappings, and they can only certify neural operators at fixed grid resolutions and provide no guarantees over the entire domain or across another resolution. We close this gap by introducing a function-space certification framework that verifies neural operators across the entire continuous domain. We formulate interval bound propagation (IBP) and CROWN in function space, propagating upper and lower bound functions through each layer using rigorous numerical integration. In addition, we extend bound propagation to higher-order derivatives to certify PDE residuals and solution error bounds. Experiments across several neural operator architectures and scientific problems, including the 572.5-million-parameter spherical Fourier neural operator for global weather forecasting, show that our method produces tight, domain-wide guarantees efficiently, providing a foundation for trustworthy operator learning.
The following analysis is generated by AI and may contain errors. It is separate from the authors’ abstract and is not peer review.
Research summary
The paper introduces a function-space certification framework for neural operators, extending IBP and CROWN to continuous domains via certified numerical integration. This rigorous approach enables domain-wide guarantees for PDE solution operators, scaling effectively to large-scale weather forecasting models and providing a foundation for trustworthy scientific machine learning.
Methodology
The paper proposes a rigorous extension of Interval Bound Propagation (IBP) and CROWN to function spaces, specifically for Fourier Neural Operators (FNOs). The core innovation is the formulation of "function-space" bound propagation, where bounds are not just scalar intervals but functions over the domain. The authors derive propagation rules for spectral convolution layers by treating them as integrals and using certified numerical integration (Gauss-Legendre and Clenshaw-Curtis quadrature) to bound the approximation error of the integral itself. This allows the verification of properties over the continuous domain rather than just discrete grid points. The method also extends to higher-order derivatives to certify PDE residuals and, via stability estimates, solution errors. The mathematical derivation is sound, bridging the gap between finite-dimensional NN verification and infinite-dimensional operator learning.
Results & evaluation
The experiments are extensive and cover a range of complexities. The authors evaluate on standard PDEs (Poisson, Helmholtz, Burgers) and chaotic dynamics (Kuramoto-Sivashinsky). A standout result is the certification of a 572.5M parameter Spherical Fourier Neural Operator (SFNO) for global weather forecasting, demonstrating that the method scales to foundation-model-sized architectures. The bounds are shown to be tight (small gap between certified and empirical bounds) and computationally efficient (hours on a single GPU). The comparison with grid-based verification highlights the necessity of the function-space approach for capturing inter-grid perturbations.
Reproducibility
The authors provide a public GitHub repository with code and datasets. The paper includes detailed algorithmic descriptions and appendices with mathematical proofs and quadrature error bounds. The use of standard libraries and clear hyperparameter settings enhances reproducibility.
Limitations
The method relies on the availability of a continuous representation of the input (e.g., spherical harmonics for weather data), which may not always be available or easy to construct. The computational cost, while manageable, grows with the quadrature resolution $Q$. The certification of adversarial robustness is limited by the exponential growth of bound looseness with network depth, so it is primarily applied to fixed-input properties like PDE residuals and regional maxima in the large-scale experiments.
Broader impact
This work is significant for the deployment of neural operators in safety-critical scientific applications (weather, fluid dynamics, structural engineering). It provides a pathway to trust data-driven PDE solvers by offering worst-case guarantees analogous to classical numerical methods. It bridges the gap between the flexibility of neural operators and the reliability required in high-stakes domains. The paper introduces a function-space certification framework for neural operators, extending IBP and CROWN to continuous domains via certified numerical integration. This rigorous approach enables domain-wide guarantees for PDE solution operators, scaling effectively to large-scale weather forecasting models and providing a foundation for trustworthy scientific machine learning.
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