Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model
arXiv:2603.17523v1 Announce Type: new Abstract: Neural Operators (NOs) are a powerful deep learning framework designed to learn the solution operator that arise from partial differential equations. This study investigates NOs ability to capture the stiff spatio-temporal dynamics of the FitzHugh-Nagumo model, which describes excitable cells. A key contribution of this work is evaluating the translation invariance using a novel training strategy. NOs are trained using an applied current with varying spatial locations and intensities at a fixed time, and the test set introduces a more challenging out-of-distribution scenario in which the applied current is translated in both time and space. This approach significantly reduces the computational cost of dataset generation. Moreover we benchmark seven NOs architectures: Convolutional Neural Operators (CNOs), Deep Operator Networks (DONs), DONs with CNN encoder (DONs-CNN), Proper Orthogonal Decomposition DONs (POD-DONs), Fourier Neural Opera
arXiv:2603.17523v1 Announce Type: new Abstract: Neural Operators (NOs) are a powerful deep learning framework designed to learn the solution operator that arise from partial differential equations. This study investigates NOs ability to capture the stiff spatio-temporal dynamics of the FitzHugh-Nagumo model, which describes excitable cells. A key contribution of this work is evaluating the translation invariance using a novel training strategy. NOs are trained using an applied current with varying spatial locations and intensities at a fixed time, and the test set introduces a more challenging out-of-distribution scenario in which the applied current is translated in both time and space. This approach significantly reduces the computational cost of dataset generation. Moreover we benchmark seven NOs architectures: Convolutional Neural Operators (CNOs), Deep Operator Networks (DONs), DONs with CNN encoder (DONs-CNN), Proper Orthogonal Decomposition DONs (POD-DONs), Fourier Neural Operators (FNOs), Tucker Tensorized FNOs (TFNOs), Localized Neural Operators (LocalNOs). We evaluated these models based on training and test accuracy, efficiency, and inference speed. Our results reveal that CNOs performs well on translated test dynamics. However, they require higher training costs, though their performance on the training set is similar to that of the other considered architectures. In contrast, FNOs achieve the lowest training error, but have the highest inference time. Regarding the translated dynamics, FNOs and their variants provide less accurate predictions. Finally, DONs and their variants demonstrate high efficiency in both training and inference, however they do not generalize well to the test set. These findings highlight the current capabilities and limitations of NOs in capturing complex ionic model dynamics and provide a comprehensive benchmark including their application to scenarios involving translated dynamics.
Executive Summary
This study evaluates the translation invariance of Neural Operators (NOs) on the FitzHugh-Nagumo model, a partial differential equation describing excitable cells. The authors propose a novel training strategy to assess NOs' ability to capture stiff spatio-temporal dynamics. Seven NOs architectures are benchmarked, and the results reveal that Convolutional Neural Operators (CNOs) perform well on translated test dynamics but require higher training costs. Fourier Neural Operators (FNOs) achieve the lowest training error but have the highest inference time. The findings highlight the capabilities and limitations of NOs in capturing complex ionic model dynamics, providing a comprehensive benchmark for future research.
Key Points
- ▸ Neural Operators (NOs) are evaluated for translation invariance on the FitzHugh-Nagumo model
- ▸ A novel training strategy is proposed to assess NOs' ability to capture stiff spatio-temporal dynamics
- ▸ Seven NOs architectures are benchmarked, including Convolutional Neural Operators (CNOs) and Fourier Neural Operators (FNOs)
Merits
Novel Training Strategy
The proposed training strategy significantly reduces the computational cost of dataset generation and enables the evaluation of NOs' translation invariance.
Benchmarking of NOs Architectures
The study provides a comprehensive benchmark of seven NOs architectures, highlighting the strengths and limitations of each.
Insights into NOs' Capabilities and Limitations
The findings provide valuable insights into the capabilities and limitations of NOs in capturing complex ionic model dynamics.
Demerits
Higher Training Costs of CNOs
CNOs require higher training costs despite achieving good performance on translated test dynamics.
Inference Time Limitations of FNOs
FNOs have the highest inference time despite achieving the lowest training error.
Limited Generalization of DONs
DONs and their variants do not generalize well to the test set despite demonstrating high efficiency.
Expert Commentary
The study presents a significant contribution to the field of deep learning for partial differential equations, providing a comprehensive benchmark of NOs architectures and their capabilities in capturing complex ionic model dynamics. The novel training strategy proposed in this study is a valuable innovation, enabling the evaluation of NOs' translation invariance with reduced computational cost. The findings highlight the strengths and limitations of each NOs architecture, providing a nuanced understanding of their capabilities. The study's implications are significant, informing the development of NOs-based models for simulating complex biological systems and informing policy decisions regarding the use of deep learning in scientific modeling and simulation.
Recommendations
- ✓ Future research should focus on developing NOs architectures that balance training cost, inference time, and generalization ability.
- ✓ The study's novel training strategy should be applied to other deep learning frameworks for partial differential equations to evaluate their translation invariance.
