This paper explores feature learning for the stationary Schrödinger equation using the deep Ritz method. It analyzes the convergence of Riemannian gradient descent, proving it reaches an approximate global minimum. The research further investigates the loss landscape when the PDE's source term follows a single-index model, demonstrating how feature alignment is achieved in both single-neuron and two-neuron scenarios. Numerical experiments validate the theory of feature emergence in the two-neuron case. AI
IMPACT This research advances techniques for solving complex differential equations, potentially impacting AI's ability to model physical systems.
RANK_REASON Academic paper detailing a novel method for solving a complex mathematical equation.
- alphaXiv
- arXivLabs
- CatalyzeX
- CORE Recommender
- DagsHub
- Deep Ritz Method for Elliptical Multiple Eigenvalue Problems
- Gotit.pub
- Hugging Face
- partial differential equation
- Ritz energy
- Schrödinger equation
- ScienceCast
- Single-index model
- two-neuron multi-index model
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