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arXiv:2609.22535

Approaching the Inverse Neutron Scattering Problem with Neural Networks

Jingyi Luo, Fletcher Williams, Ang-Kun Wu, David Dahlbom, Cristian D. Batista, Hao Zhang

Published Sep 18, 2026• 0 citations
arXivHTMLPDF

Abstract

Determining the microscopic Hamiltonian of a quantum magnet from inelastic neutron scattering measurements remains a central challenge in condensed matter physics. While the dynamical spin structure factor contains, in principle, sufficient information about the underlying interactions, extracting Hamiltonian parameters from experimental spectra constitutes a highly nontrivial inverse problem due to the large parameter space and the presence of experimental noise. Here we demonstrate that neural networks can efficiently solve this inverse problem for a broad class of quantum magnets. As a benchmark, we consider an eight-parameter family of honeycomb lattice Heisenberg spin models in the fully polarized phase, where the dynamical spin structure factor can be computed exactly within linear spin-wave theory. Using a large synthetic dataset of neutron scattering spectra, we train neural networks to infer the underlying Hamiltonian parameters directly from the dynamical response. We compare the performance of three different architectures---fully connected neural networks (FCNNs), one-dimensional convolutional neural networks (CNN1Ds), and two-dimensional convolutional neural networks (CNN2Ds)---and find that all achieve high predictive accuracy. Remarkably, the trained models remain robust in the presence of substantial experimental uncertainty, reliably recovering the Hamiltonian parameters even when the input spectra are contaminated by random noise with amplitudes reaching $10\%$ of the signal intensity. Our results establish machine learning as a powerful framework for quantitative Hamiltonian reconstruction from neutron scattering data and provide a practical route toward automated characterization of quantum magnetic materials.

DOI
10.48550/arxiv.2609.22535

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Determining the microscopic Hamiltonian of a quantum magnet from inelastic neutron scattering measurements remains a central challenge in condensed matter physics. While the dynamical spin structure factor contains, in principle, sufficient information about the underlying interactions, extracting Hamiltonian parameters from experimental spectra constitutes a highly nontrivial inverse problem due to the large parameter space and the presence of experimental noise. Here we demonstrate that neural networks can efficiently solve this inverse problem for a broad class of quantum magnets. As a benchmark, we consider an eight-parameter family of honeycomb lattice Heisenberg spin models in the fully polarized phase, where the dynamical spin structure factor can be computed exactly within linear spin-wave theory. Using a large synthetic dataset of neutron scattering spectra, we train neural networks to infer the underlying Hamiltonian parameters directly from the dynamical response. We compare the performance of three different architectures---fully connected neural networks (FCNNs), one-dimensional convolutional neural networks (CNN1Ds), and two-dimensional convolutional neural networks (CNN2Ds)---and find that all achieve high predictive accuracy. Remarkably, the trained models remain robust in the presence of substantial experimental uncertainty, reliably recovering the Hamiltonian parameters even when the input spectra are contaminated by random noise with amplitudes reaching 10%10\%10% of the signal intensity. Our results establish machine learning as a powerful framework for quantitative Hamiltonian reconstruction from neutron scattering data and provide a practical route toward automated characterization of quantum magnetic materials.
Hamiltonian learning (HL), enabling precise estimation of system parameters and underlying dynamics, plays a critical role in characterizing quantum systems. However, conventional HL methods face challenges in noise robustness and resource efficiency, especially under limited measurements. In this work, we present Inverse Physics-Informed Neural Networks for Hamiltonian Learning (iPINN-HL), an approach that incorporates the Schrödinger equation as a soft constraint via a loss function penalty into the ML procedure. This formulation allows the model to integrate both observational data and known physical laws to infer Hamiltonian parameters with greater accuracy and resource efficiency. We benchmark iPINN-HL against a deep-neural-network-based quantum state tomography method [denoted as DNN-HL (Deep Neural Network for Hamiltonian Learning)] and demonstrate its effectiveness across several different scenarios, including one-dimensional spin chains, cross-resonance gate calibration, crosstalk identification, and real-time compensation to parameter drift. Our results show that iPINN-HL can approach the Heisenberg limit and exhibits robustness to noises, while outperforming DNN-HL in accuracy and resource efficiency. Therefore, iPINN-HL is a powerful and flexible framework for quantum system characterization for practical tasks.
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The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.
We state and prove two theorems about the ground state of magnetic systems described by very general Heisenberg-type models and discuss their implications for magnetic neutron scattering. The first theorem states that two models cannot have the same correlator without sharing the corresponding ground states. According to the second theorem, an NNN-qubit wave function cannot reproduce the correlators of a given system unless it represents a true ground state of that system. We discuss the implications for neutron scattering inverse problems. We argue that the first theorem provides a framework to understand neutron-based Hamiltonian learning. Furthermore, we propose a variational approach to quantum magnets based on the second theorem in which a representation of the wave function (held, for instance, in a neural network or in the qubit register of a quantum processor) is optimized to fit experimental neutron scattering data directly, without the involvement of a model Hamiltonian.
Materials design is an inverse problem: finding the couplings of a microscopic Hamiltonian whose ground state exhibits a desired property. For strongly correlated quantum systems, this remains largely out of reach, as each candidate Hamiltonian requires its own costly simulation. Here we show that Foundation Neural-Network Quantum States, single wave functions that encode the ground states of an entire family of Hamiltonians, make this search efficient. Because the Hamiltonian couplings enter as neural-network inputs, any ground-state property can be optimized by gradient descent directly in coupling space. We use this approach to search for nonmagnetic phases in two-dimensional spin models in coupling spaces of increasing dimensionality. Specifically, we focus on quantum spin liquids, exotic phases of matter in which quantum fluctuations prevent magnetic order at zero temperature. In the largest search space considered, spanned by eight superexchange couplings and containing many paradigmatic frustrated spin systems, the optimization converges to the J1J_1J1​-J2J_2J2​-δδδ Heisenberg model when starting from a magnetic phase. We show that the ground state at the optimum set of couplings has neither magnetic nor valence-bond order in the thermodynamic limit, leading to a new quantum spin-liquid candidate. The approach is completely general and can be extended to other models and physical properties to optimize, demonstrating that Neural-Network Quantum States are tools not only for simulating quantum matter but also for designing it.