AI 中文总结
研究一维非线性双曲平衡定律，提出物理信息令牌变换器（PITT）方法，结合多种技术。通过在史瓦西 - 伯格斯方程上测试，与有限体积近似比较，分析伯格斯极限，展示激波位置渐近传播定律，适用于相关复杂动力学问题。
AI 中文摘要
我们为一维非线性双曲平衡定律引入了一种物理信息令牌变换器（PITT）方法，使用分段稳态剖面来表示近似弱解。该方法结合了符号方程令牌化、傅里叶神经算子编码器、用于激波运动的显式兰金 - 于戈尼奥定律以及一个学习校正项。为清晰起见，我们针对相对论性史瓦西 - 伯格斯方程（史瓦西背景下球对称流体流动的标量模型）进行阐述。对于此模型，稳态不变量和广义黎曼解是明确的，可纳入神经演化。特别是，主导间断由解析跳跃条件推进，学习部分重建平滑区域、稀疏波、几何依赖性和有限分辨率效应。该方法旨在精确定位波前并保持相关稳态。我们在移动激波、驻定激波、稀疏波上测试了PITT方法，并与标准高阶有限体积近似进行比较。我们还分析了标准伯格斯极限（当史瓦西质量趋于零时）。兰金 - 于戈尼奥先验在这些测试中起主导作用，而方程令牌化带来系统的额外增益。该方法适用于涉及几何效应和/或复杂激波动力学的问题，在此用于研究稳态解扰动的长期动力学。特别是，我们展示了扰动稳态流激波位置的渐近传播定律。
英文摘要
We introduce a Physics-Informed Token Transformer (PITT) methodology for nonlinear hyperbolic balance laws in one space dimension, using piecewise steady-state profiles for the representation of approximate weak solutions. The method combines symbolic equation tokenization, a Fourier neural operator encoder, an explicit Rankine--Hugoniot law for shock motion, and a learned correction term. For clarity, we present it here for the relativistic Schwarzschild--Burgers equation, a scalar model for spherically symmetric fluid flows on a Schwarzschild background. For this model the steady-state invariant and the generalized Riemann solutions are explicit, and they can therefore be built into the neural evolution. In particular, the leading discontinuities are advanced by the analytical jump condition, while the learned part reconstructs smooth regions, rarefaction fans, geometric dependence, and finite-resolution effects. The method is designed to locate wave fronts accurately and to preserve the relevant steady states. We test our PITT method on moving shocks, stationary shocks, rarefaction waves, and compare it with a standard high-order finite-volume approximation. We also analyze the standard Burgers limit (when the Schwarzschild mass tends to zero). The Rankine--Hugoniot prior plays the dominant role in these tests, while equation tokenization gives a systematic additional gain. The method is relevant for problems involving geometric effects and/or complex shock-wave dynamics, and is used here to study the long-time dynamics of perturbations of steady-state solutions. In particular, we exhibit an asymptotic law of propagation for the shock location of perturbed steady-state flows.
Comments34 pages
