mlsanityData to 14 Oct 2026

Generative Neural Operators through Diffusion Last Layer

arXiv 2602.04139 ↗First posted 4 Feb 2026cs.LG physics.comp-phcited by 0

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Neural operators provide a powerful framework for learning discretization invariant mappings between function spaces, but standard deterministic models do not capture predictive uncertainty. We introduce diffusion last layer (DLL), a modular probabilistic output head for neural operator backbones. DLL represents target fields through an input dependent low rank expansion inspired by the Karhunen-Lo'eve expansion and learns a conditional diffusion model over the corresponding coefficient space. This design enables efficient distributional modeling while preserving the structural advantages of operator learning. On stochastic PDE benchmarks with random forcing, DLL achieves strong distributional fidelity and performs competitively with pixel space and conventional latent diffusion baselines. In deterministic long horizon rollout tasks, DLL improves rollout stability over the underlying backbone and provides useful estimates of predictive uncertainty under compounding autoregressive errors. These results suggest that diffusion modeling in learned coefficient spaces offers a practical route to uncertainty aware neural operators.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 66 of its 79 references that are in the map. First posted on arXiv 4 Feb 2026.

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66 of 79 references.

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Other references · 55

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
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  3. 2017On Calibration of Modern Neural Networkscited by 4,960
  4. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  5. 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
  6. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  7. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  8. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  9. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
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  7. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  8. 2021Factorized Fourier Neural Operatorscited by 249
  9. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 233
  10. 2020Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networkscited by 231
  11. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  12. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 216
  13. 2024Poseidon: Efficient Foundation Models for PDEscited by 200
  14. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  15. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 152
  16. 2023HyperDiffusion: Generating Implicit Neural Fields with Weight-Space Diffusioncited by 152
  17. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 151
  18. 2019Convolutional Conditional Neural Processescited by 142
  19. 2023Score-based Data Assimilationcited by 96
  20. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
  21. 2024Variational Bayesian Last Layerscited by 69
  22. 2023Functional Flow Matchingcited by 67
  23. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 66
  24. 2024On conditional diffusion models for PDE simulationscited by 60
  25. 2024APEBench: A Benchmark for Autoregressive Neural Emulators of PDEscited by 60
  26. 2022Generative Adversarial Neural Operatorscited by 52
  27. 2022Neural Diffusion Processescited by 50
  28. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 47
  29. 2021The Gaussian Neural Processcited by 34
  30. 2024Wavelet Diffusion Neural Operatorcited by 29
  31. 2025Generative Latent Neural PDE Solver using Flow Matchingcited by 28
  32. 2024Calibrated Uncertainty Quantification for Operator Learning via Conformal Predictioncited by 26
  33. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 25
  34. 2024Text2PDE: Latent Diffusion Models for Accessible Physics Simulationcited by 25
  35. 2025Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulationcited by 19
  36. 2025Probabilistic neural operators for functional uncertainty quantificationcited by 18
  37. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
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