🤖 AI Summary
This study addresses the numerical instability of neural spectral architectures (NeuSA) when solving stiff differential equations by proposing a novel framework that integrates neural spectral methods with exponential time differencing (ETD). The approach leverages spectral representations and high-order exponential integrators to precisely handle linear stiff terms, while employing physics-informed neural networks to model nonlinear residuals, thereby achieving efficient decoupling of stiff–nonstiff dynamics. Experimental results demonstrate that the proposed framework attains both high stability and accuracy on stiff PDE benchmarks. Furthermore, it supports the identification of unknown physical parameters through inverse problem learning, offering a reliable new paradigm for modeling stiff systems.
📝 Abstract
Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and lack of causality. Neuro-Spectral Architectures (NeuSA), a recently proposed alternative to PINNs, mitigate both issues, but their numerical integration becomes unstable for stiff differential equations arising in many relevant physical problems. This study proposes Neuro-Spectral Exponential Time Differencing Architectures (NEXT), which combines the spectral representation of the PDE solution in NeuSA with high-order exponential integrators. Within this approach, the linear stiff part of the vector field induced by the PDE is integrated exactly through matrix exponentials, while the possibly nonlinear remainder is modeled by a neural network. The effectiveness of NEXT is verified through benchmark experiments on a set of stiff PDEs, in which NEXT is stable and accurate while NeuSA diverges numerically. It is also shown that NEXT can be applied to inverse problems, where the model has to learn unknown parameters or boundary conditions from sparse data. All code used in this work is publicly available at: https://github.com/marcioh2m/next.git .