We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end of the spectrum. We present several deep network architectures that predict the localization of bounded states from a sample of a potential. For tackling higher dimensional problems, we consider a class of physics-informed deep dense networks. In particular, we focus on the interpretability of the proposed approaches. Deep network is used as a general reduced order model that describes the nonlinear connection between the potential and the ground state. The performance of the surrogate reduced model is controlled by an error estimator and the model is updated if necessary. Finally, we present a host of experiments to measure the accuracy and performance of the proposed algorithm.

中文翻译：

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用于逼近由限制势定位的特征模式的深度神经网络模型

我们研究了一类椭圆反应扩散算子的本征模式定位。作为原型模型问题，我们使用由随机势参数化的薛定谔哈密顿量族并研究相关的有效约束势。这个问题是在有限域中提出的，我们在谱的下端计算局部有界状态。我们提出了几种深度网络架构，可以从潜在的样本中预测有界状态的定位。为了解决更高维度的问题，我们考虑了一类基于物理的深度密集网络。我们特别关注所提出方法的可解释性。深度网络用作描述电位和基态之间非线性连接的通用降阶模型。代理简化模型的性能由误差估计器控制，并且在必要时更新模型。最后，我们提出了大量实验来衡量所提出算法的准确性和性能。