mlsanityData to 7 Oct 2026

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

arXiv 2607.06369 ↗First posted 7 Jul 2026math.OC stat.MLcited by 0

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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.

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

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References · 40

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  4. 2022Neural Networks can Learn Representations with Gradient Descentcited by 161
  5. 2022High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representationcited by 146
  6. 2023SGD learning on neural networks: leap complexity and saddle-to-saddle dynamicscited by 142
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  8. 2019On the Power and Limitations of Random Features for Understanding Neural Networkscited by 103
  9. 2022Learning Single-Index Models with Shallow Neural Networkscited by 92
  10. 2023How Two-Layer Neural Networks Learn, One (Giant) Step at a Timecited by 67
  11. 2023Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Modelscited by 60
  12. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
28 more
  1. 2017Learning ReLUs via Gradient Descentcited by 54
  2. 2020Agnostic Learning of a Single Neuron with Gradient Descentcited by 46
  3. 2023A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networkscited by 46
  4. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  5. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  6. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  7. 2024Asymptotics of feature learning in two-layer networks after one gradient-stepcited by 35
  8. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  9. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  10. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  11. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  12. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 18
  13. 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16
  14. 2023Agnostically Learning Single-Index Models using Omnipredictorscited by 16
  15. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  16. 2025Survey on Algorithms for multi-index modelscited by 15
  17. 2020A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networkscited by 13
  18. 2022Learning a Single Neuron for Non-monotonic Activation Functionscited by 13
  19. 2022Agnostic Learning of General ReLU Activation Using Gradient Descentcited by 12
  20. 2023Robustly Learning a Single Neuron via Sharpnesscited by 12
  21. 2024Robustly Learning Single-Index Models via Alignment Sharpnesscited by 11
  22. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
  23. 2019Deep neural network approximations for Monte Carlo algorithmscited by 6
  24. 2024Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networkscited by 5
  25. 2024DRM Revisited: A Complete Error Analysiscited by 4
  26. 2024Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Modelscited by 4
  27. 2023Numerical solution of Poisson partial differential equations in high dimension using two-layer neural networkscited by 1
  28. 2024Two-layers neural networks for Schr"odinger eigenvalue problemscited by 1

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