ASPEN: An Adaptive Spectral Physics-Enabled Network for Ginzburg-Landau Dynamics

TL;DR

ASPEN is a novel neural network architecture that adaptively learns spectral features to accurately solve complex, nonlinear PDEs like the Ginzburg-Landau equation, outperforming standard PINNs especially on stiff, multi-scale problems.

Contribution

The paper introduces ASPEN, an adaptive spectral layer integrated into PINNs, enabling dynamic spectral basis learning to effectively solve challenging nonlinear PDEs.

Findings

ASPEN accurately solves the Ginzburg-Landau equation with low residuals.

Standard PINNs fail on the same problem, diverging into non-physical oscillations.

ASPEN captures physical properties like energy relaxation and domain stability.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful, mesh-free paradigm for solving partial differential equations (PDEs). However, they notoriously struggle with stiff, multi-scale, and nonlinear systems due to the inherent spectral bias of standard multilayer perceptron (MLP) architectures, which prevents them from adequately representing high-frequency components. In this work, we introduce the Adaptive Spectral Physics-Enabled Network (ASPEN), a novel architecture designed to overcome this critical limitation. ASPEN integrates an adaptive spectral layer with learnable Fourier features directly into the network's input stage. This mechanism allows the model to dynamically tune its own spectral basis during training, enabling it to efficiently learn and represent the precise frequency content required by the solution. We demonstrate the efficacy of ASPEN by applying it to the complex Ginzburg-Landau equation (CGLE), a canonical and challenging benchmark for nonlinear, stiff spatio-temporal dynamics. Our results show that a standard PINN architecture catastrophically fails on this problem, diverging into non-physical oscillations. In contrast, ASPEN successfully solves the CGLE with exceptional accuracy. The predicted solution is visually indistinguishable from the high-resolution ground truth, achieving a low median physics residual of 5.10 x 10^-3. Furthermore, we validate that ASPEN's solution is not only pointwise accurate but also physically consistent, correctly capturing emergent physical properties, including the rapid free energy relaxation and the long-term stability of the domain wall front. This work demonstrates that by incorporating an adaptive spectral basis, our framework provides a robust and physically-consistent solver for complex dynamical systems where standard PINNs fail, opening new options for machine learning in challenging physical domains.