TASE-Stabilized Time-Evolving Natural Gradient Methods for Diffusion-Dominated PDEs
Abstract
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 Tp(hW0) to each Runge--Kutta stage increment before projection, where W0 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.
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BibTeX
@misc{shi2026tasestabilized,
title = {{TASE-Stabilized Time-Evolving Natural Gradient Methods for Diffusion-Dominated PDEs}},
author = {Zihao Shi and Dongling Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23054},
eprint = {2609.23054},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23054},
}
RIS
TY - UNPB
TI - TASE-Stabilized Time-Evolving Natural Gradient Methods for Diffusion-Dominated PDEs
AU - Zihao Shi
AU - Dongling Wang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23054
UR - https://arxiv.org/abs/2609.23054
AB - 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.
ER -