用一般索伯列夫空间的谱逼近学习偏微分方程
Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
摘要：
利用切比雪夫多项式给出了一般索伯列夫空间的一种新的有限维谱逼近.基于这个多项式代理模型（PSM），我们实现了一个变分公式，解决了大量的线性和非线性偏微分方程（PDE）。PSM与物理信息神经网络（PINN）一样灵活，并为解决逆PDE问题（如PDE参数推断）提供了一种替代方案。与PINN相比，PSM导致大量PDE（包括所有线性PDE）的凸优化问题，在这种情况下，由于潜在的变分梯度下降的指数收敛速度，PSM近似是可有效计算的。作为一个实际的结果，突出的PDE问题解决了PSM没有高性能计算（HPC）在本地机器上。这种效率上的提高得到了近似能力的提高，在精度和运行时间上都优于PINN替代方案。除了我们在这里给出的经验证据之外，经典PDE理论在索伯列夫空间近似下的翻译表明PSM普遍适用于适定的正则正逆PDE问题。
英文摘要：
We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent. As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime. Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.
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