A Neural Network for the Identical Kuramoto Equation: Architectural Considerations and Performance Evaluation
Nishantak Panigrahi, Mayank Patwal
TL;DR
This work applies physics-informed neural networks to a nonlocal conservation law derived from the identical Kuramoto model, aiming to approximate the phase-density solution efficiently. By systematically varying activation functions, network depth/width, and training strategies, the authors reveal that moderately sized tanh networks achieve stable convergence with competitive accuracy, while sine activations can offer marginal gains at the risk of artefacts and ReLU fails for this problem. A key finding is the intrinsic oversmoothing of discontinuities by standard feed-forward PINNs, demonstrated with singular and piecewise-constant initial data, and supported by experiments on energy-norm errors and training times. The paper provides actionable guidelines for practitioners and discusses potential remedies—such as variational losses, domain decomposition, and Fourier features—to extend PINNs to discontinuous or highly nonlocal PDEs, thereby informing future development in neural scientific computing.
Abstract
In this paper, we investigate the efficiency of Deep Neural Networks (DNNs) to approximate the solution of a nonlocal conservation law derived from the identical-oscillator Kuramoto model, focusing on the evaluation of an architectural choice and its impact on solution accuracy based on the energy norm and computation time. Through systematic experimentation, we demonstrate that network configuration parameters-specifically, activation function selection (tanh vs. sin vs. ReLU), network depth (4-8 hidden layers), width (64-256 neurons), and training methodology (collocation points, epoch count)-significantly influence convergence characteristics. We observe that tanh activation yields stable convergence across configurations, whereas sine activation can attain marginally lower errors and training times in isolated cases, but occasionally produce nonphysical artefacts. Our comparative analysis with traditional numerical methods shows that optimally configured DNNs offer competitive accuracy with notably different computational trade-offs. Furthermore, we identify fundamental limitations of standard feed-forward architectures when handling singular or piecewise-constant solutions, providing empirical evidence that such networks inherently oversmooth sharp features due to the natural function space limitations of standard activation functions. This work contributes to the growing body of research on neural network-based scientific computing by providing practitioners with empirical guidelines for DNN implementation while illuminating fundamental theoretical constraints that must be overcome to expand their applicability to more challenging physical systems with discontinuities.