New proof tightens discretization error bounds for SSM neural operators
Researchers bridge theory and practice for neural PDE solvers with formal stability guarantees.
Neural operators like DeepONet and Fourier Neural Operators (FNOs) have revolutionized solving partial differential equations (PDEs) by learning mappings between function spaces. However, a persistent gap remains between their continuous mathematical foundations and the discrete data used in practice—raising questions about numerical stability and error propagation. This new paper from Bendahi, Fradin, Peralez, Digne, and Nadri (arXiv:2605.18905) directly addresses that gap by establishing rigorous theoretical guarantees for discretization error and stability in neural operator schemes.
The authors derive analytical bounds that tie solution regularity to input discretization, providing a formal quantification of accuracy under real-world numerical constraints. They specialize these bounds to State Space Model-based Neural Operators (SS-NOs) and FNOs, proving a new discretization error theorem. Additionally, an input-to-state stability (ISS) analysis formally assesses how discretization affects results obtained in the continuous domain. Empirical experiments on 1D and 2D benchmarks (e.g., Darcy flow) validate the theoretical bounds and demonstrate SS-NOs' robustness across varying grid resolutions. This work offers a critical step toward making neural operators reliable for engineering and scientific computing applications.
- Derived discretization error theorem linking solution regularity to input discretization for SS-NOs and FNOs.
- Input-to-state stability analysis formally assesses impact of discretization on continuous-domain results.
- Empirical validation on 1D and 2D PDE benchmarks confirms theoretical bounds and robustness.
Why It Matters
Bridges critical gap between continuous neural operator theory and practical discrete implementation for reliable PDE solving.