用神经网络逼近参数依赖问题的解，方法简单且收敛有保障。
Approximation of solutions of parameter-dependent problems by residual neural networks
- 基于梯度流训练含解析激活函数的神经网络
- 能准确捕捉微分方程解对少数参数的依赖关系
- 适合求解严重不适定的反问题，尤其在困难区域仍有效
我们提出一种基于梯度流训练含解析激活函数神经网络的收敛算法，其收敛性由Lojasiewicz理论保证。该方法实现简单，通过求解常微分方程组来逼近网络系数。实验验证了残差神经网络对参数化问题解的逼近能力：简单常微分方程解对少量参数的依赖关系被正确再现；即使在严重不适定的波约束反问题中，也能合理近似解，包括困难区域。
原文摘要 · Abstract (English)
We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.
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