Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
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| Format: | Preprint |
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2025
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| author | Behroozi, Abdolmehdi and, Chaopeng Shen Kifer, Daniel |
| author_facet | Behroozi, Abdolmehdi and, Chaopeng Shen Kifer, Daniel |
| contents | 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 |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations Behroozi, Abdolmehdi and, Chaopeng Shen Kifer, Daniel Machine Learning Computational Engineering, Finance, and Science 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 |
| title | Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations |
| topic | Machine Learning Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2505.08740 |