mlsanityData to 14 Oct 2026

Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations

arXiv 2505.08740 ↗First posted 13 May 2025cs.LG cs.CEcited by 14

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Abstract (arXiv)

Parametric differential equations of the form du/dt = f(u, x, t, p) are fundamental in science and engineering. While deep learning frameworks such as the Fourier Neural Operator (FNO) can efficiently approximate solutions, they struggle with inverse problems, sensitivity estimation (du/dp), and concept drift. We address these limitations by introducing a sensitivity-based regularization strategy, called Sensitivity-Constrained Fourier Neural Operators (SC-FNO). SC-FNO achieves high accuracy in predicting solution paths and consistently outperforms standard FNO and FNO with physics-informed regularization. It improves performance in parameter inversion tasks, scales to high-dimensional parameter spaces (tested with up to 82 parameters), and reduces both data and training requirements. These gains are achieved with a modest increase in training time (30% to 130% per epoch) and generalize across various types of differential equations and neural operators. Code and selected experiments are available at: https://github.com/AMBehroozi/SC_Neural_Operators

Key facts. Cited by 14 papers in the map (data through 14 Oct 2026). Builds on 18 of its 40 references that are in the map. First posted on arXiv 13 May 2025.

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18 of 40 references.

References · 18

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  4. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  5. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  6. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 234
  7. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 233
  8. 2017Sobolev Training for Neural Networkscited by 143
  9. 2018Auxiliary Tasks in Multi-task Learningcited by 108
  10. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  11. 2019A Differentiable Programming System to Bridge Machine Learning and Scientific Computingcited by 83
  12. 2020Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flowscited by 37
6 more
  1. 2020From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modelingcited by 26
  2. 2022Differentiable, learnable, regionalized process-based models with physical outputs can approach state-of-the-art hydrologic prediction accuracycited by 21
  3. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  4. 2022Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEscited by 13
  5. 2021Physics-constrained deep neural network method for estimating parameters in a redox flow batterycited by 9
  6. 2023Differentiable Modeling and Optimization of Battery Electrolyte Mixtures Using Geometric Deep Learningcited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 11.

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  1. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  2. 2025FNOPE: Simulation-based inference on function spaces with Fourier Neural Operatorscited by 4
  3. 2025Generalized Spherical Neural Operators: Green's Function Formulationcited by 3
  4. 2026Spectral Audit of In-Context Operator Networkscited by 2
  5. 2026When Does AI for PDEs Yield Scientific Evidence?cited by 1
  6. 2026KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operatorscited by 1
  7. 2025Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurementscited by 0
  8. 2026Neural Operators for Biomedical Spherical Heterogeneitycited by 0
  9. 2026Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEscited by 0
  10. 2026Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Errorcited by 0
  11. 2026Frequency Selective Neural Networks as a Foundation Architecture for Time Series Learningcited by 0
  12. 2026Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domainscited by 0
2 more
  1. 2026Derivative-Informed Training of Neural Operators On-the-Fly via Sketched Tangent Consistencycited by 0
  2. 2026LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Predictioncited by 0