Fast and Reliable Probabilistic Reflectometry Inversion with Prior-Amortized Neural Posterior Estimation

TL;DR

The paper tackles the phaseless inverse problem in X-ray and neutron reflectometry by introducing PANPE, a prior-amortized neural posterior estimator that delivers fast, reliable, and multimodal Bayesian inference for thin-film structures. By combining simulation-based neural posterior estimation with adaptive, physics-informed priors and a GPU-accelerated transfer-matrix simulator, PANPE achieves coverage of the true posterior and refines estimates with likelihood-based methods, enabling real-time or high-throughput analysis. The approach demonstrates multimodal posteriors on synthetic data, accurate marginal posteriors on experimental XRR data, and effective co-refinement of neutron measurements, underscoring the importance of priors and equivariances in inverse scattering. With its adaptable priors, adaptive $q$-discretization, and capability to guide experimental decisions, PANPE offers a scalable, generalizable framework for fast, probabilistic reflectometry analysis and other complex inverse problems.

Abstract

Reconstructing the structure of thin films and multilayers from measurements of scattered X-rays or neutrons is key to progress in physics, chemistry, and biology. However, finding all structures compatible with reflectometry data is computationally prohibitive for standard algorithms, which typically results in unreliable analysis with only a single potential solution identified. We address this lack of reliability with a probabilistic deep learning method that identifies all realistic structures in seconds, setting new standards in reflectometry. Our method, Prior-Amortized Neural Posterior Estimation (PANPE), combines simulation-based inference with novel adaptive priors that inform the inference network about known structural properties and controllable experimental conditions. PANPE networks support key scenarios such as high-throughput sample characterization, real-time monitoring of evolving structures, or the co-refinement of several experimental data sets, and can be adapted to provide fast, reliable, and flexible inference across many other inverse problems.

Figures (6)

• Figure 1: Reflectometry analysis. (a) A schematic experimental setup for reflectometry measurements. The reflected intensity $R(q)$ from a studied layered structure as a function of momentum transfer $q$ contains information about parameters $\theta$ of the studied sample. The momentum transfer is typically controlled by the geometry in X-ray reflectometry or by the energy in time-of-flight neutron measurements. (b) Inverse problem in reflectometry: the forward simulations provided by the scattering theory should be inverted during inference, which is generally ambiguous. (c) Inference methods commonly employed for reflectometry analysis, as well as our proposed approach. The standard maximum likelihood estimation approach provides a single solution by design. MCMC locally explores the parameter space and can overlook distributional modes. In contrast, (PA)NPE posterior estimate is guaranteed to cover all the solutions, with further refinement based on likelihood evaluation improving accuracy. Our prior amortization method, PANPE, enables the analysis of multiple experimental scenarios using a single neural network.
• Figure 2: Multimodal posterior distribution obtained by PANPE-IS on a simulated reflectivity curve with 10 free parameters for a two-layer structure. The neural network produces results in accordance with the provided prior information, identifying (a) multiple solutions for a "wide" prior distribution and (c) a single distributional mode for a "narrow" prior (gray dashed lines). Colors denote distinct distributional modes obtained by clustering samples. The corresponding reflectivity curves (b) and (d) enable real-time likelihood-based refinement, resulting in accurate posterior estimation. The corner plot (e) shows the resulting marginalized 4d distributions obtained for both priors along with the colored samples related to the colored profiles in (a).
• Figure 3: Sample efficiencies for conventional importance sampling (left) and our PANPE-IS model (right) on a test dataset of 1000 simulated curves (blue) and a experimental dataset of 208 X-ray reflectometry curves (orange). An additional axis on the right-hand side indicates the estimated time it takes to generate 100 effective samples (ESS) on our hardware with the efficient GPU-accelerated reflectometry simulator (see text). Both the simulated and experimental data consist of two-layered structures with 10 parameters, but in the experimental data, only the top layer is unknown, as the parameters of the silicon substrate and the silicon oxide layer are largely constrained through their respective priors.
• Figure 4: Marginal distributions of the thickness $d_1$, roughness $\sigma_1$, and density $\rho_1$ of the diindenoperylene (DIP) layer growing on a silicon substrate for three in situ experimental XRR datasets obtained by our model (on the left) and via conventional importance sampling from prior distribution (on the right). The colors designate normalized probability densities. Purple dots correspond to conventional manual fits performed via differential evolution reported in pithan2022reflectometry_data. SM Figure 4 shows time-dependent sample efficiencies and the log evidence estimations for both methods.
• Figure 5: Experimental XRR curve analyzed using both a wide prior distribution that encompasses the entire training range (shown in red) and a narrow, physics-informed prior distribution (shown in blue). (a) SLD profiles associated with the PANPE-IS samples. Profiles with the highest likelihood are highlighted with bold lines. (b) The observed reflectivity curve (in gray) is compared with simulated curves that correspond to the maximum likelihood from both the narrow and wide prior distributions. While both fits are satisfactory, the unphysical solution (in red) has a likelihood that is more than $10^6$ times greater than its physical counterpart due to larger residuals (c). In this case, when trained solely with a wide prior distribution, the NPE network mainly samples unphysical solutions, which appear much more probable without the physics-informed prior. The use of prior amortization addresses this issue.