Deep learning-based approach for modeling threshold curves

The initiation of a propagating wave in a one-dimensional excitable medium depends on a time-independent critical solution, whose stable manifold defines the threshold curve separating initial conditions that lead to propagation wave solutions from those that ultimately decay. Accurately identifying this threshold curve is particularly important for understanding the dynamics of excitable media. Hence, there have been some numerical and analytical approximations of these threshold curves, yet capturing them precisely remains difficult due to the nonlinear and complex nature of these systems. This study introduces Physics-Informed Neural Networks (PINNs) to model threshold curve in the excitable media, using the Zeldovich–Frank–Kamenetsky equation (also known as Nagumo equation) as a test case. PINNs integrate physical principles with machine learning to achieve high accuracy and efficiency in capturing the behavior of the threshold curve, offering a promising alternative to traditional methods. To improve computational efficiency, transfer learning is applied, allowing pre-trained model parameters to adapt quickly and effectively to new scenarios. We validate the accuracy of the PINN-predicted threshold curve through comparisons with direct numerical simulations and the analytical strength-extent curve, showing strong alignment with traditional numerical results and theoretical predictions.