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

2025/05/13 by Abdolmehdi Behroozi, Behroozi, Abdolmehdi, Chaopeng Shen and +3 · 2 citations
Computer Science · Mathematics · #Computational Engineering #FOS: Computer and information sciences #Finance #Machine Learning (cs.LG) #Neural Networks and Applications #Numerical methods in inverse problems #Statistical and numerical algorithms #and Science (cs.CE)

paper · pdf · doi:10.48550/arxiv.2505.08740

openalex publication_date 2025/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Differentiable Modeling and Optimization of Battery Electrolyte Mixtures Using Geometric Deep Learning
• Idealized Baroclinic Wave Test Results
• Physics-Informed Neural Operator for Learning Partial Differential Equations
• Multiwavelet-based Operator Learning for Differential Equations
• Physics-constrained deep neural network method for estimating parameters in a redox flow battery
• Physics-informed machine learning
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• Fourier Neural Operator for Parametric Partial Differential Equations
• Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows
• NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
• A Differentiable Programming System to Bridge Machine Learning and Scientific Computing
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Neural Ordinary Differential Equations
• Auxiliary Tasks in Multi-task Learning
• Why So Many Published Sensitivity Analyses Are False. A Systematic Review of Sensitivity Analysis Practices
• Sobolev Training for Neural Networks
• On the usefulness of overparameterized ecological models

Cited by

• Generalized Spherical Neural Operators: Green's Function Formulation
• Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements
• FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators

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