[2505.18288] Operator Learning for Schrödinger Equation: Unitarity, Error Bounds, and Time Generalization

Statistics > Machine Learning arXiv:2505.18288 (stat) [Submitted on 23 May 2025 (v1), last revised 3 Apr 2026 (this version, v2)] Title:Operator Learning for Schrödinger Equation: Unitarity, Error Bounds, and Time Generalization Authors:Yash Patel, Unique Subedi, Ambuj Tewari View a PDF of the paper titled Operator Learning for Schr\"{o}dinger Equation: Unitarity, Error Bounds, and Time Generalization, by Yash Patel and 2 other authors View PDF HTML (experimental) Abstract:We consider the problem of learning the evolution operator for the time-dependent Schrödinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the Schrödinger 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-...