# Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method

arXiv: https://arxiv.org/abs/2607.06369 · HTML: https://mlsanity.com/paper/2607.06369/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 60 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 30 of 40)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [How Two-Layer Neural Networks Learn, One (Giant) Step at a Time](https://mlsanity.com/paper/2305.18270/) (2023)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Learning ReLUs via Gradient Descent](https://mlsanity.com/paper/1705.04591/) (2017)
- [A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks](https://mlsanity.com/paper/2310.07891/) (2023)
- [Agnostic Learning of a Single Neuron with Gradient Descent](https://mlsanity.com/paper/2005.14426/) (2020)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Asymptotics of feature learning in two-layer networks after one gradient-step](https://mlsanity.com/paper/2402.04980/) (2024)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Agnostically Learning Single-Index Models using Omnipredictors](https://mlsanity.com/paper/2306.10615/) (2023)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Survey on Algorithms for multi-index models](https://mlsanity.com/paper/2504.05426/) (2025)
- [A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks](https://mlsanity.com/paper/2001.06145/) (2020)
- [Learning a Single Neuron for Non-monotonic Activation Functions](https://mlsanity.com/paper/2202.08064/) (2022)

## Abstract (arXiv)

This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schr"odinger 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 ε of 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.