Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

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.

Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators

Ionic models, described by systems of stiff ordinary differential equations, are fundamental tools for simulating the complex dynamics of excitable cells in both Computational Neuroscience and Cardiology. Approximating these models using Artificial Neural Networks poses significant challenges due to their inherent stiffness, multiscale nonlinearities, and the wide range of dynamical behaviors they exhibit, including multiple equilibrium points, limit cycles, and intricate interactions. While in previous studies the dynamics of the transmembrane potential has been predicted in low dimensionality settings, in the present study we extend these results by investigating whether Fourier Neural Operators can effectively learn the evolution of all the state variables within these dynamical systems in higher dimensions. We demonstrate the effectiveness of this approach by accurately learning the dynamics of three well-established ionic models with increasing dimensionality: the two-variable FitzHugh-Nagumo model, the four-variable Hodgkin-Huxley model, and the forty-one-variable O'Hara-Rudy model. To ensure the selection of near-optimal configurations for the Fourier Neural Operator, we conducted automatic hyperparameter tuning under two scenarios: an unconstrained setting, where the number of trainable parameters is not limited, and a constrained case with a fixed number of trainable parameters. Both constrained and unconstrained architectures achieve comparable results in terms of accuracy across all the models considered. However, the unconstrained architecture required approximately half the number of training epochs to achieve similar error levels, as evidenced by the loss function values recorded during training. These results underline the capabilities of Fourier Neural Operators to accurately capture complex multiscale dynamics, even in high-dimensional dynamical systems.