Ever since the discovery of x-rays, considerable breakthroughs have been made using them as a probe of matter, from testing models of the atom to solving the structure of deoxyribonucleic acid (DNA). Over the last few decades with the proliferation of synchrotron x-ray sources around the world, the application to many scientific fields has progressed tremendously and allowed studies of complicated structures and phenomena like protein dynamics and crystallography [1, 2], electronic structures of strongly correlated materials [3, 4], and a wide variety of elementary excitations [5, 6]. With the the development of the next generation of light sources, especially the x-ray free electron lasers (X-FEL) [7, 8], not only have discoveries accelerated, but completely novel techniques have been developed and new fields of science have emerged, such as laboratory astrophysics [9, 10, 11, 12] and single particle diffractive imaging [13, 14, 15]. Among these emerging techniques brought by X-FELs, the development of x-ray photon fluctuation spectroscopy (XPFS) holds particular relevance for condensed matter and material physics [16]. XPFS is a unique and powerful approach that opens up numerous opportunities to probe ultrafast dynamics of timescales corresponding to the µeV to meV-energy level. As the high-level coherence of the x-ray beam encodes subtle changes in the system at these timescales, XPFS is capable of investigating fluctuations of elementary excitations, such as that of the spin [17]. The fluctuation spectra collected using this method can be directly related back to correlation functions derived from Hamiltonians [18, 19], yielding invaluable experimental insights for theoretical developments and deeper understandings of the underlying physics.

The observation and description of collective excitations in solids is a fundamental issue when seeking to understand the physics of a many-body system. Analysis of these excitations is usually carried out by measuring the dynamical structure factor, S(Q, $\omega$), with inelastic neutron or x-ray scattering techniques and comparing this against a calculated dynamical model. Here, we develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. We benchmark this approach on a Linear Spin Wave Theory (LSWT) simulator and advanced inelastic neutron scattering data from the square-lattice spin-1 antiferromagnet La$_2$NiO$_4$. We find that the model predicts the unknown parameters with excellent agreement relative to analytical fitting. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data, without the need for human-guided peak finding and fitting algorithms. This prototypical approach promises a new technology for this field to automatically detect and refine more advanced models for ordered quantum systems.