Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
Abdolmehdi Behroozi Chaopeng Shen Daniel Kifer
Parametric differential equations of the form  $\frac{\partial u}{\partial t} = f(u, x, t, p)$ are fundamental in science and engineering. While deep learning frameworks like the Fourier Neural Operator (FNO) efficiently approximate differential equation solutions, they struggle with inverse problems, sensitivity calculations  $\frac{\partial u}{\partial p}$, and concept drift.  We address these challenges by introducing a novel sensitivity loss regularizer, demonstrated through Sensitivity-Constrained Fourier Neural Operators (SC-FNO). Our approach maintains high accuracy for solution paths and outperforms both standard FNO and FNO with Physics-Informed Neural Network regularization. SC-FNO exhibits superior performance in parameter inversion tasks, accommodates more complex parameter spaces (tested with up to 82 parameters), reduces training data requirements, and decreases training time while maintaining accuracy.  These improvements apply across various differential equations and neural operators, enhancing their reliability without significant computational overhead (30%–130% extra training time per epoch). Models and selected experiment code are available at:  [https://github.com/AMBehroozi/SC_Neural_Operators](https://github.com/AMBehroozi/SC_Neural_Operators).
Fourier Neural Operator, Sensitivity Analysis, Parametric Differential Equations, Surrogate Modes, Differentiable Numerical Solvers, Inverse Problems
Introducing a novel Sensitivity-Constrained Fourier Neural Operator (SC-FNO) to improve the accuracy of parametric differential equation solutions and their sensitivities using differentiable numerical solvers.