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

arXiv: https://arxiv.org/abs/2505.08740 · HTML: https://mlsanity.com/paper/2505.08740/

## 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.

## Builds on (most cited first, 18 of 18)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Auxiliary Tasks in Multi-task Learning](https://mlsanity.com/paper/1805.06334/) (2018)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows](https://mlsanity.com/paper/2005.13420/) (2020)
- [From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling](https://mlsanity.com/paper/2007.15751/) (2020)
- [Differentiable, learnable, regionalized process-based models with physical outputs can approach state-of-the-art hydrologic prediction accuracy](https://mlsanity.com/paper/2203.14827/) (2022)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEs](https://mlsanity.com/paper/2204.01205/) (2022)
- [Physics-constrained deep neural network method for estimating parameters in a redox flow battery](https://mlsanity.com/paper/2106.11451/) (2021)
- [Differentiable Modeling and Optimization of Battery Electrolyte Mixtures Using Geometric Deep Learning](https://mlsanity.com/paper/2310.03047/) (2023)

## Built on by (most cited first, 14 of 14)

- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)
- [Generalized Spherical Neural Operators: Green's Function Formulation](https://mlsanity.com/paper/2512.10723/) (2025)
- [Spectral Audit of In-Context Operator Networks](https://mlsanity.com/paper/2606.02427/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operators](https://mlsanity.com/paper/2609.27216/) (2026)
- [Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements](https://mlsanity.com/paper/2509.00203/) (2025)
- [Neural Operators for Biomedical Spherical Heterogeneity](https://mlsanity.com/paper/2601.03561/) (2026)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)
- [Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Error](https://mlsanity.com/paper/2608.15373/) (2026)
- [Frequency Selective Neural Networks as a Foundation Architecture for Time Series Learning](https://mlsanity.com/paper/2608.29012/) (2026)
- [Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains](https://mlsanity.com/paper/2608.29892/) (2026)
- [Derivative-Informed Training of Neural Operators On-the-Fly via Sketched Tangent Consistency](https://mlsanity.com/paper/2609.32797/) (2026)
- [LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Prediction](https://mlsanity.com/paper/2609.34493/) (2026)

## 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