mlsanityData to 7 Oct 2026

Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learning

arXiv 2507.10401 ↗First posted 10 Jul 2025cs.LG math.PRcited by 3

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Abstract (arXiv)

We develop a novel framework for uncertainty quantification in operator learning, the Stochastic Operator Network (SON). SON combines the stochastic optimal control concepts of the Stochastic Neural Network (SNN) with the DeepONet. By formulating the branch net as an SDE and backpropagating through the adjoint BSDE, we replace the gradient of the loss function with the gradient of the Hamiltonian from Stohastic Maximum Principle in the SGD update. This allows SON to learn the uncertainty present in operators through its diffusion parameters. We then demonstrate the effectiveness of SON when replicating several noisy operators in 2D and 3D.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 19 of its 39 references that are in the map. First posted on arXiv 10 Jul 2025.

Builds on

19 of 39 references.

References · 19

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2016Optimization Methods for Large-Scale Machine Learningcited by 1,140
  5. 2019Augmented Neural ODEscited by 396
  6. 2017Stable Architectures for Deep Neural Networkscited by 340
  7. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  8. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  9. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 168
  10. 2019Neural Jump Stochastic Differential Equationscited by 147
  11. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  12. 2019Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noisecited by 98
7 more
  1. 2020SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimatescited by 75
  2. 2022Generative Adversarial Neural Operatorscited by 51
  3. 2023On the training and generalization of deep operator networkscited by 31
  4. 2018Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networkscited by 20
  5. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  6. 2017Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Datacited by 17
  7. 2024A Mathematical Analysis of Neural Operator Behaviorscited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equationscited by 4
  2. 2026Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equationscited by 1
  3. 2026Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorizationcited by 0