mlsanityData to 14 Oct 2026

Operator Learning at Machine Precision

arXiv 2511.19980 ↗First posted 25 Nov 2025cs.LG cs.NA math.NAcited by 7

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Abstract (arXiv)

Neural operator learning methods have garnered significant attention in scientific computing for their ability to approximate infinite-dimensional operators. However, increasing their complexity often fails to substantially improve their accuracy, leaving them on par with much simpler approaches such as kernel methods and more traditional reduced-order models. In this article, we set out to address this shortcoming and introduce CHONKNORIS (Cholesky Newton--Kantorovich Neural Operator Residual Iterative System), an operator learning paradigm that can achieve machine precision. CHONKNORIS draws on numerical analysis: many nonlinear forward and inverse PDE problems are solvable by Newton-type methods. Rather than regressing the solution operator itself, our method regresses the Cholesky factors of the elliptic operator associated with Tikhonov-regularized Newton--Kantorovich updates. The resulting unrolled iteration yields a neural architecture whose machine-precision behavior follows from achieving a contractive map, requiring far lower accuracy than end-to-end approximation of the solution operator. We benchmark CHONKNORIS on a range of nonlinear forward and inverse problems, including a nonlinear elliptic equation, Burgers' equation, a nonlinear Darcy flow problem, the Calder'on problem, an inverse wave scattering problem, and a problem from seismic imaging. We also present theoretical guarantees for the convergence of CHONKNORIS in terms of the accuracy of the emulated Cholesky factors. Additionally, we introduce a foundation model variant, FONKNORIS (Foundation Newton--Kantorovich Neural Operator Residual Iterative System), which aggregates multiple pre-trained CHONKNORIS experts for diverse PDEs to emulate the solution map of a novel nonlinear PDE. Our FONKNORIS model is able to accurately solve unseen nonlinear PDEs such as the Klein--Gordon and Sine--Gordon equations.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 43 of its 75 references that are in the map. First posted on arXiv 25 Nov 2025.

Builds on

43 of 75 references.

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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 6.

7 papers.

  1. 2025Convolutional-neural-operator-based transfer learning for solving PDEscited by 2
  2. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  3. 2026Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennescited by 0
  4. 2026Learning Contractive Integral Operators with Fredholm Integral Neural Operatorscited by 0
  5. 2026Kernel Methods for Learning Operators with Multiple Inputs and Outputscited by 0
  6. 2026A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solverscited by 0
  7. 2026Transferability of Learned States in Neural PDE Solverscited by 0