随机权重神经网络可高效逼近非线性偏微分方程解
Random Neural Network Expressivity for Non-Linear Partial Differential Equations
- 用随机初始化隐藏层权重的神经网络逼近非线性PDE解
- 对光滑函数实现无维度依赖的1/2阶逼近率
- 适用于多孔介质方程和可压缩纳维-斯托克斯方程
具有随机生成隐藏权重的神经网络（RaNN）在实际应用中广泛使用，既作为独立学习方法，也作为全可训练深度学习模型的初始化。本文研究了RaNN在求解非线性偏微分方程（PDEs）时的表达能力。尽管其广泛应用，但关于其逼近性质的严格理论理解仍有限。我们推导了时间依赖Sobolev函数的误差界，对足够光滑的函数获得维度无关的1/2阶逼近率。将结果应用于两类重要非线性PDE：多孔介质方程与可压缩纳维-斯托克斯方程，证明了RaNN能高效逼近这些复杂非线性方程的解。理论分析得到数值实验支持，表明收敛速率可推广至更广设置。
原文摘要 · Abstract (English)
Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.
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