Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
This work addresses challenges in solving parametric differential equations for science and engineering applications, offering an incremental improvement over existing neural operator methods.
The paper tackles the limitations of Fourier Neural Operators (FNOs) in handling inverse problems, sensitivity estimation, and concept drift for parametric differential equations by introducing Sensitivity-Constrained Fourier Neural Operators (SC-FNO), which improves accuracy in solution prediction and parameter inversion, scales to high-dimensional spaces (up to 82 parameters), and reduces data and training requirements with a modest increase in training time (30% to 130% per epoch).
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