用神经网络求解二维周期势中的量子本征问题，无需网格和标注数据。
Physics-Informed Neural Solvers for Periodic Quantum Eigenproblems
- 用神经网络联合学习布洛赫波函数与能量，通过物理约束构建损失函数。
- 在布里渊区内训练可准确还原石墨烯类蜂窝晶格的能带结构。
- 支持迁移学习，可从弱势场快速适应强势场下的拓扑变化。
本文提出一种物理信息神经网络框架，用于求解二维周期势中粒子的弗洛凯-布洛赫本征问题，重点研究具有狄拉克点拓扑特性的蜂窝晶格结构，该结构在石墨烯等材料中具有重要意义。通过神经网络同时学习复杂的布洛赫函数及其对应本征值（能量），构建无网格求解器，通过复合损失函数强制满足薛定谔方程、布洛赫周期性及归一化约束，无需监督信号。模型在布里渊区内进行训练，以恢复能带结构和布洛赫模式，并与传统的平面波展开法进行数值验证。进一步探索了迁移学习技术，使求解器可从近自由电子势快速适配至强变势场，展现其捕捉能带拓扑变化的能力。本工作为量子本征问题的物理信息机器学习方法提供了新思路，揭示了对称性、能带结构与神经网络架构之间的相互作用。
原文摘要 · Abstract (English)
This thesis presents a physics-informed machine learning framework for solving the Floquet-Bloch eigenvalue problem associated with particles in two-dimensional periodic potentials, with a focus on honeycomb lattice geometry, due to its distinctive band topology featuring Dirac points and its relevance to materials such as graphene. By leveraging neural networks to learn complex Bloch functions and their associated eigenvalues (energies) simultaneously, we develop a mesh-free solver enforcing the governing Schrödinger equation, Bloch periodicity, and normalization constraints through a composite loss function without supervision. The model is trained over the Brillouin zone to recover band structures and Bloch modes, with numerical validation against traditional plane-wave expansion methods. We further explore transfer learning techniques to adapt the solver from nearly-free electron potentials to strongly varying potentials, demonstrating its ability to capture changes in band structure topology. This work contributes to the growing field of physics-informed machine learning for quantum eigenproblems, providing insights into the interplay between symmetry, band structure, and neural architectures.
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