AI 中文总结
针对扩散主导PDE的刚性稳定性问题，提出TASE-TENG方法，利用可学习的TASE稳定算子改进显式龙格-库塔投影，提升稳定性和精度，并支持不同分辨率重建。
AI 中文摘要
时间演化自然梯度（TENG）方法通过局部自然梯度迭代将时间离散的目标状态投影到神经网络流形上，从而演化时间相关偏微分方程的神经近似。然而，对于刚性扩散主导的问题，显式龙格-库塔方法面临严重的稳定性限制。高频扩散分量可能将阶段目标驱动到神经流形的局部可达区域之外，恶化局部最小二乘投影的条件数，并使参数更新失稳。我们提出TASE-TENG，该方法在投影之前将高度稳定的显式算子$T_p(hW_0)$应用于每个龙格-库塔阶段增量，其中$W_0$近似主导的刚性扩散算子。为了在无网格设置中构造该替代算子，我们开发了一个可学习算子，其局部交互系数由共享的几何依赖神经规则生成。结构化分解强制满足离散自伴性、耗散性和常数保持性。该算子离线训练，在线积分期间保持固定，仅用于TASE稳定算子内部。在刚性扩散和反应-扩散问题上的数值实验表明，在相同时间步长下，稳定性和精度得到改善，高频扩散模式引起的误差减少。学习到的规则还允许在不重新训练的情况下，以不同的空间点分辨率重建稳定算子。
英文摘要
The Time-Evolving Natural Gradient (TENG) method evolves neural approximations of time-dependent partial differential equations by projecting time-discrete target states onto the neural-network manifold through local natural-gradient iterations. For stiff diffusion-dominated problems, however, explicit Runge--Kutta methods face severe stability restrictions. High-frequency diffusive components can drive stage targets beyond the locally reachable region of the neural manifold, worsening the conditioning of local least-squares projections and destabilizing parameter updates. We propose TASE--TENG, which applies the highly stable explicit operator $T_p(hW_0)$ to each Runge--Kutta stage increment before projection, where $W_0$ approximates the dominant stiff diffusion operator. To construct this surrogate in a mesh-free setting, we develop a learnable operator whose local interaction coefficients are generated by a shared geometry-dependent neural rule. A structured factorization enforces discrete self-adjointness, dissipativity, and constant preservation. The operator is trained offline and remains fixed during online integration, where it is used only within the TASE stabilization operator. Numerical experiments on stiff diffusion and reaction--diffusion problems demonstrate improved stability and accuracy at the same time-step size, with reduced errors from high-frequency diffusive modes. The learned rule also allows the stabilization operator to be reconstructed at different spatial point resolutions without retraining.
