# Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalization

arXiv: https://arxiv.org/abs/2505.18288 · HTML: https://mlsanity.com/paper/2505.18288/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 66 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 24 of 24)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [A neural operator-based surrogate solver for free-form electromagnetic inverse design](https://mlsanity.com/paper/2302.01934/) (2023)
- [Learning Unitary Operators with Help From u(n)](https://mlsanity.com/paper/1607.04903/) (2016)
- [Scattering with Neural Operators](https://mlsanity.com/paper/2308.14789/) (2023)
- [Ab-initio variational wave functions for the time-dependent many-electron Schr"odinger equation](https://mlsanity.com/paper/2403.07447/) (2024)
- [Deep learning and the Schr"odinger equation](https://mlsanity.com/paper/1702.01361/) (2017)
- [On the Benefits of Active Data Collection in Operator Learning](https://mlsanity.com/paper/2410.19725/) (2024)
- [Learning S-Matrix Phases with Neural Operators](https://mlsanity.com/paper/2404.14551/) (2024)
- [Partially Unitary Learning](https://mlsanity.com/paper/2405.10263/) (2024)
- [Neural Quantum Propagators for Driven-Dissipative Quantum Dynamics](https://mlsanity.com/paper/2410.16091/) (2024)
- [Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems](https://mlsanity.com/paper/2409.03302/) (2024)

## Abstract (arXiv)

We consider the problem of learning the evolution operator for the time-dependent Schr"odinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the Schr"odinger equation, such as linearity and unitarity, and lack theoretical guarantees on prediction error or time generalization. To address this, we introduce a linear estimator for the evolution operator that preserves a weak form of unitarity. We establish both upper bounds and lower bounds on the prediction error of the proposed estimator that hold uniformly over classes of sufficiently smooth initial wave functions. Additionally, we derive time generalization bounds that quantify how the estimator extrapolates beyond the time points seen during training. Experiments across real-world Hamiltonians -- including hydrogen atoms, ion traps for qubit design, and optical lattices -- show that our estimator achieves relative errors up to two orders of magnitude smaller than state-of-the-art methods such as the Fourier Neural Operator and DeepONet.