Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
Abdolmehdi Behroozi, Chaopeng Shen and, Daniel Kifer
TL;DR
This paper introduces SC-FNO, a regularization method for Fourier Neural Operators that enhances their ability to solve forward and inverse parametric differential equations, especially in high-dimensional parameter spaces.
Contribution
The paper proposes a sensitivity-based regularization strategy, SC-FNO, which improves the accuracy, scalability, and efficiency of neural operators for parametric differential equations.
Findings
SC-FNO outperforms standard FNO in solution prediction accuracy.
It significantly improves parameter inversion capabilities.
The method scales effectively to high-dimensional parameter spaces, up to 82 parameters.
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…
Peer Reviews
Decision·ICLR 2025 Poster
- This paper examines a notable case involving the system parameter $P$, where $ \frac{du}{dp} $ is computed as a sensitivity loss to regularize the model. - The results indicate that SC-FNO achieves greater accuracy in $ \frac{du}{dp} $ and demonstrates enhanced robustness against perturbations in $p$. - This approach aids inverse modeling in identifying the system parameters more effectively.
### Writing The overall writing quality could be enhanced, with several important details suggested to be moved from the appendix into the main text. - The physical parameter $P$ is central to the study and should be introduced more thoroughly. Currently, it is presented in a general form, but it’s important to clarify whether $P$ is a scalar in $\mathbb{R}$, a vector in $\mathbb{R}^d$, or a function in $L(\mathbb{R})$. - Starting with a concrete example in the introduction would improve clarity
The study addresses the limitations observed when Fourier neural operators, combined with a physics-informed loss function, may yield suboptimal performance. The proposed regularization method appears to mitigate these challenges effectively.
The proposed approach requires additional information and computational resources, as it leverages direct solution-based information, such as a precomputed Jacobian in the differentiable numerical solver (Eq, (6)). Another concern is the regularizer’s potential weakness to noise in practical settings. Derivative-based techniques are typically vulnerable to data noise, which could affect prediction accuracy, especially in applications beyond simulation, such as parameter inversion. In real-worl
1. The paper's proposed Sensitivity-Constrained approach is applicable to various neural operators, offering high research value. 2. The paper is clearly written, with no redundant mathematical derivations, and offers excellent readability. 3. In modeling PDEs, this approach not only enables accurate approximation of the target variable but also effectively models its derivatives.
1. There is no enough discussion on the memory overhead introduced by AD (Automatic Differentiation) and FD (Finite Differences). 2. The experiments are limited. It is not sure whether this method remains effective in complicated and high-resolution PDE scenarios.
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Taxonomy
TopicsNeural Networks and Applications · Numerical methods in inverse problems · Statistical and numerical algorithms