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

• URL: http://arxiv.org/abs/2505.08740v3
• Date: Sat, 31 May 2025 04:11:52 GMT
• Title: Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
• Authors: Abdolmehdi Behroozi, Chaopeng Shen and, Daniel Kifer,
• Abstract summary: Parametric differential equations of the form du/dt = f(u, x, t, p) are fundamental in science and engineering.<n>Deep learning frameworks such as the Fourier Neural Operator (FNO) can efficiently approximate solutions, but struggle with inverse problems, sensitivity estimation (du/dp), and concept drift.<n>We address these limitations by introducing a sensitivity-based regularization strategy, called Sensitivity-Constrained Fourier Neural Operators (SC-FNO)<n>SC-FNO achieves high accuracy in predicting solution paths and consistently outperforms standard FNO and FNO with physics-informed regularization.
• Score: 6.900101619562999
• License: http://creativecommons.org/licenses/by/4.0/
• 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

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