基于深度里兹法的高维稳态薛定谔方程的特征学习
Feature Learning for the High Dimensional Stationary Schödinger Equation with Deep Ritz Method
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中文总结 AI 辅助
研究基于深度里兹法求解高维稳态薛定谔方程的特征学习，分析了黎曼梯度下降收敛性，探讨了源项为单指标模型时不同模型的损失景观，通过数值实验验证双神经元设置下的特征出现理论。
中文摘要 AI 辅助
本文研究了在具有诺伊曼边界条件的稳态薛定谔方程求解中，深度里兹法框架内的特征学习。首先在不可知环境下分析了黎曼梯度下降的收敛性，证明经过特定迭代次数后损失在最优损失常数倍的\(\epsilon\)范围内。接着研究了偏微分方程源项为单指标模型时的损失景观，分析了单指标模型和双神经元多指标模型的情况。最后通过数值实验验证了双神经元设置下的特征出现理论。
英文摘要
This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first analyze the convergence of Riemannian gradient descent in an agnostic setting, where the hypothesis function is restricted to a single-index model while the PDE solution is arbitrary. We prove that gradient descent reaches an approximate global minimum: after T = O(log(1/ε)) iterations, the loss is within εof a constant multiple of the optimal loss. We then examine the loss landscape when the source term of the PDE itself follows a single-index model, considering hypothesis functions given by either a single-index model or a two-neuron multi-index model. In the single-index case, we show that the minimum Ritz energy is attained at the feature vector aligned with that of the source term. In the two-neuron case, we study the landscape of regularized Ritz losses and characterize how a second feature emerges, given that the first feature is aligned with the source, as the regularization parameter varies. Finally, numerical experiments are presented to validate the feature emergence theory in the two-neuron setting.
