NEXT 架构结合指数积分器，解决刚性 PDE 数值发散问题
NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
PINN 解刚性 PDE 容易翻车，这篇把指数积分器接进 NeuSA，稳定性和精度都保住了，代码也开源了。
arXiv 论文提出 NEXT（Neuro-Spectral Exponential Time Differencing Architectures），在 NeuSA 的谱表示基础上引入高阶指数积分器，用矩阵指数精确积分 PDE 引导向量场中的线性刚性部分，非线性余项交由神经网络建模。在多个刚性 PDE 基准实验中，NEXT 保持稳定且准确，而 NeuSA 数值发散。论文还展示了 NEXT 在反问题上的应用，可从稀疏数据学习未知参数或边界条件。代码已在 GitHub 公开。
NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
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 .