We provide a numerical analysis and computation of neural network projected schemes for approximating one dimensional Wasserstein gradient flows. We approximate the Lagrangian mapping functions of gradient flows by the class of two-layer neural network functions with ReLU (rectified linear unit) activation functions. The numerical scheme is based on a projected gradient method, namely the Wasserstein natural gradient, where the projection is constructed from the $L^2$ mapping spaces onto the neural network parameterized mapping space. We establish theoretical guarantees for the performance of the neural projected dynamics. We derive a closed-form update for the scheme with well-posedness and explicit consistency guarantee for a particular choice of network structure. General truncation error analysis is also established on the basis of the projective nature of the dynamics. Numerical examples, including gradient drift Fokker-Planck equations, porous medium equations, and Keller-Segel models, verify the accuracy and effectiveness of the proposed neural projected algorithm.

翻译：本文对用于近似一维Wasserstein梯度流的神经网络投影格式进行了数值分析与计算。我们利用具有ReLU（修正线性单元）激活函数的两层神经网络函数类来近似梯度流的拉格朗日映射函数。该数值格式基于投影梯度方法，即Wasserstein自然梯度法，其中投影由$L^2$映射空间到神经网络参数化映射空间构造而成。我们为神经投影动力学的性能建立了理论保证。针对特定网络结构，推导了具有适定性和显式一致性保证的格式闭式更新。基于动力学的投影性质，还建立了通用截断误差分析。包括梯度漂移Fokker-Planck方程、多孔介质方程和Keller-Segel模型在内的数值算例验证了所提出的神经投影算法的准确性与有效性。