Compositional neural networks for multimodal X-ray and neutron scattering inversion

Valentin Munteanu; Alexander Hinderhofer; Dmitry Lapkin; Vladimir Starostin; Frank Schreiber

cs.LGOct 21, 2026

Compositional neural networks for multimodal X-ray and neutron scattering inversion

Authors: Valentin Munteanu, Alexander Hinderhofer, Dmitry Lapkin, Vladimir Starostin, Frank Schreiber

Organizations: Institute of Applied Physics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany · Cluster of Excellence ”Machine Learning” - new perspectives for science, University of Tübingen, Germany

Abstract

X-ray and neutron scattering are indispensable for structural characterization across condensed-matter physics and materials science. Yet, because scattering experiments generally record intensities without phase information, distinct structures can yield similar scattering patterns, rendering the corresponding inverse problems non-unique. This ambiguity can be reduced by combining complementary measurements, but their integration into machine-learning frameworks remains challenging. Here we introduce a modular framework for probabilistic multi-measurement analysis based on prior-amortized flow matching. Single-observation posterior estimators are trained separately for individual observation types or physical probes and combined only at inference time through a compositional sampler. This late fusion approach allows pretrained single-observation models to be reused without dedicated joint retraining. As a case in point, we consider reflectometry, which provides depth-resolved, nanometer-scale information about thin films and interfaces. Our single-observation demonstrations span X-ray, time-of-flight neutron and polarized neutron reflectometry. In separate co-refinement studies, task-matched estimators enable homogeneous composition across multiple neutron contrasts and heterogeneous multimodal composition of X-ray and neutron measurements. Together, these results establish compositional flow matching as a modular approach to uncertainty-aware inversion and co-refinement, with potential applicability to other scattering problems in which multiple observations constrain a common physical structure while retaining measurement-specific parameters.

Figures & tables

Figure 1: Prior-amortized flow matching posterior estimation and compositional co-refinement for reflectometry. (a) Schematic X-ray reflectometry (XRR) geometry. The reflectivity

R_{\mathrm{XRR}}

is measured by varying the incident angle

\alpha_{i}

at fixed wavelength

\lambda

, giving access to

R_{\mathrm{XRR}}(q(\alpha_{i},\lambda))

. (b) Schematic time-of-flight neutron reflectometry (TOF-NR) geometry. The incident angle is fixed, while the wavelength dependence is resolved from neutron time of flight, giving

R_{\mathrm{NR}}(q(\alpha_{i},\lambda))

. (c) Compositional flow matching posterior estimation. Each observation

y_{j}

, together with the corresponding prior bounds

b_{j}

, defines a conditioning context

C_{j}

. A single-observation flow matching model maps the relevant latent coordinates

\Theta_{t}^{(j)}=(\theta_{t}^{\mathrm{sh}},\theta^{\mathrm{loc}}_{j,t})

to a local vector-field contribution. The local contributions are embedded into a common shared/local parameter space, combined into a composed vector field, and integrated from a normal base distribution to obtain samples from the joint posterior over the global parameter vector

\Theta=(\theta^{\mathrm{sh}},\theta^{\mathrm{loc}}_{1},\ldots,\theta^{\mathrm{loc}}_{n})

. (d) Schematic two-dimensional posterior representation for parameters

\theta_{i}

and

\theta_{j}

. The dashed lines indicate the user-supplied prior bounds, and the blue region denotes posterior density retained within the selected bounded parameter domain. Grey regions indicate density outside the selected bounds that is excluded by the bounded prior. (e) Posterior parameter samples can be propagated through the physical sample parameterization to obtain depth-dependent SLD profiles. In multimodal XRR/NR co-refinement, shared structural parameters determine the common layer geometry, while probe-specific local parameters determine the electronic SLD profile

\rho_{e}(z)

and the nuclear SLD profile

\rho_{n}(z)

Figure 2: Single-curve posterior inference for XRR, TOF-NR and polymer-brush NR data. (a) Experimental XRR curve for a one-layer PTCDI-C 3 film on a Si substrate, shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. (b) SLD profiles for the PTCDI-C 3 film reconstructed from the FMPE-MCMC posterior. (c) Experimental TOF-NR curve for a two-layer PS:Bis-PCBM/Bis-PCBM structure on Si, with data from Ref. [ 31 ] , shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. (d) SLD profiles reconstructed from the FMPE-MCMC posterior for the TOF-NR data. (e) Experimental NR curve for the polymer-brush structure, with data from Ref. [ 37 ] , shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. (f) Solvent volume-fraction profiles reconstructed from the FMPE-MCMC posterior for the polymer-brush structure.

Figure 3: Multichannel posterior inference for full spin-resolved PNR and compositional co-refinement of multiple-contrast NR data. (a) Synthetic full spin-resolved PNR data, shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. The curves corresponding to different spin channels are shifted vertically for clarity. (b) Nuclear and magnetic SLD profiles reconstructed from the FMPE-MCMC posterior. (c) Magnetization-angle profiles reconstructed from the FMPE-MCMC posterior. (d) Experimental NR curves corresponding to three solvent contrasts, D 2 O, H 2 O and CM, with data from Ref. [ 55 ] , shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. The curves are shifted vertically for clarity. (e) SLD profiles for the three solvent contrasts reconstructed from the FMPE-MCMC posterior.

Figure 4: Multimodal compositional co-refinement of XRR and PNR data. (a) Experimental XRR and PNR data for the Pt/Fe structure, with data from Ref. [ 37 ] , shown together with posterior-predictive simulations generated from FMPE-MCMC posterior samples. (b) Reconstructed real and imaginary electronic SLD profiles. (c) Reconstructed nuclear and magnetic SLD profiles.

Figure 5: Neural network architecture. The reflectivity data is first mapped to a fixed-dimensional representation by an embedding network. This embedding is processed by a residual MLP with GELU activations and layer-normalized residual blocks. Conditioning is applied through FiLM layers using

t

\theta_{t}

, the prior bounds

b

, and optional auxiliary variables

\eta

. For the neutron-reflectometry models considered here,

\eta

includes the linear smearing coefficient

dq/q

. The network outputs the conditional vector field

v(\theta_{t},t;C)

Figure 6: Experimental XRR data for a one-layer PTCDI-C 3 film on a Si substrate are shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 7: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the XRR data of the PTCDI-C 3 film. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , SLDs $\rho_{i}$ , the $q$ -shift offset $q_{\mathrm{sh}}$ and the reflectivity scale factor $r_{\mathrm{sc}}$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , $\rho_{i}$ in $10^{-6}\,\mathrm{\AA}^{-2}$ , $q_{\mathrm{sh}}$ in $\mathrm{\AA}^{-1}$ , and $r_{\mathrm{sc}}$ is dimensionless. The subscript B denotes the backing medium. The plot ranges correspond to the prior bounds.

Figure 8: Experimental XRR data for a one-layer C 60 film on a Si substrate are shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 9: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the XRR data of the C 60 film. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , SLDs $\rho_{i}$ , the $q$ -shift offset $q_{\mathrm{sh}}$ and the reflectivity scale factor $r_{\mathrm{sc}}$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , $\rho_{i}$ in $10^{-6}\,\mathrm{\AA}^{-2}$ , $q_{\mathrm{sh}}$ in $\mathrm{\AA}^{-1}$ , and $r_{\mathrm{sc}}$ is dimensionless. The subscript B denotes the backing medium. The plot ranges correspond to the prior bounds.

Figure 10: Experimental TOF-NR data for the PS:Bis-PCBM/Bis-PCBM structure, shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 11: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the NR data from the time-of-flight reflectometer. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , SLDs $\rho_{i}$ , the reflectivity scale factor $r_{\mathrm{sc}}$ and the $\log_{10}$ background. Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , $\rho_{i}$ in $10^{-6}\,\mathrm{\AA}^{-2}$ , while $r_{\mathrm{sc}}$ and the $\log_{10}$ background are dimensionless. The subscript B denotes the backing medium. The plot ranges correspond to the prior bounds.

Figure 12: Experimental NR data for the polymer-brush structure, shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 13: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the NR data of the polymer-brush structure (data sourced from [ 37 ] ). Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the polymer volume fraction at the grafting interface $\phi_{0}$ , the length of the brush $L$ , the polymer-brush exponent $n$ , the thickness of the reactive layer $d_{\mathrm{rl}}$ , the SLD of the polymer $\rho_{\mathrm{pol}}$ , the roughness of the substrate $\sigma_{\mathrm{sub}}$ and the roughness of the reactive layer $\sigma_{\mathrm{rl}}$ . Units: $L$ , $d_{\mathrm{rl}}$ , $\sigma_{\mathrm{sub}}$ and $\sigma_{\mathrm{rl}}$ are in $\mathrm{\AA}$ , $\rho_{\mathrm{pol}}$ is in $10^{-6}\,\mathrm{\AA}^{-2}$ , and $\phi_{0}$ and $n$ are dimensionless. The plot ranges correspond to the prior bounds.

Figure 14: Synthetic full spin-resolved PNR data, shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 15: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the synthetic full spin-resolved PNR data. The displayed parameters comprise layer thicknesses $d_{i}$ , nuclear SLDs $\rho^{n}_{i}$ , magnetic SLDs $\rho^{m}_{i}$ and magnetization angles $\psi^{m}_{i}$ . Units: $d_{i}$ are in $\mathrm{\AA}$ , $\rho^{n}_{i}$ and $\rho^{m}_{i}$ are in $10^{-6}\,\mathrm{\AA}^{-2}$ , and $\psi^{m}_{i}$ is in degrees. The subscript B denotes the backing medium. The plot ranges correspond to the prior bounds.

Figure 16: Experimental NR data corresponding to three solvent contrasts (D 2 O, H 2 O and CM), shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 17: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the three-contrast NR data. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the layer thicknesses $d_{i}$ , the SLD of the polymer $\rho_{2}$ without the solvent, the solvent volume fraction in the polymer layer $\phi^{\mathrm{solv}}_{2}$ and the reflectivity scale factors $r_{\mathrm{sc}}$ and $\log_{10}$ backgrounds for each curve. Units: $d_{i}$ are in $\mathrm{\AA}$ , $\rho_{2}$ is in $10^{-6}\,\mathrm{\AA}^{-2}$ , and $\phi^{\mathrm{solv}}_{2}$ , $r_{\mathrm{sc}}$ and the $\log_{10}$ backgrounds are dimensionless. The plot ranges correspond to the prior bounds.

Figure 18: Experimental XRR and PNR data for the Pt/Fe structure, shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 19: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the Pt/Fe structure. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , the reflectivity scale factors $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and, for PNR, the mixing coefficient $\alpha$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , while $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and $\alpha$ are dimensionless. The plot ranges correspond to the prior bounds.

Figure 20: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior, in the case of an alternative inference where the SLDs are not fixed, for the Pt/Fe structure. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , real electronic SLDs, imaginary electronic SLDs, nuclear SLDs, magnetic SLDs, reflectivity scale factors $r_{\mathrm{sc}}$ , $\log_{10}$ backgrounds and, for PNR, the mixing coefficient $\alpha$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , all SLDs are in $10^{-6}\,\mathrm{\AA}^{-2}$ , while $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and $\alpha$ are dimensionless. The plot ranges correspond to the prior bounds.

Figure 21: Experimental XRR and PNR data for the Pt/Co structure, shown together with posterior-predictive simulations generated from parameter samples. (a) Posterior-predictive curves obtained directly from the raw FMPE posterior. (b) Posterior-predictive curves obtained after FMPE-initialized MCMC refinement.

Figure 22: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior inferred for the Pt/Co structure. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , the magnetic SLD of the Co layer $\rho^{m}_{2}$ , the reflectivity scale factors $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and, for PNR, the mixing coefficient $\alpha$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , $\rho^{m}_{2}$ is in $10^{-6}\,\mathrm{\AA}^{-2}$ , while $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and $\alpha$ are dimensionless. The plot ranges correspond to the prior bounds.

Figure 23: Corner plot of the raw FMPE (green) and FMPE-MCMC (red) posterior, in the case of an alternative inference where the SLDs are not fixed, for the Pt/Co structure. Filled contours show the 68% and 95% credible regions. The displayed parameters comprise the layer thicknesses $d_{i}$ , interfacial roughnesses $\sigma_{i}$ , real electronic SLDs, imaginary electronic SLDs, nuclear SLDs, magnetic SLDs, reflectivity scale factors $r_{\mathrm{sc}}$ , $\log_{10}$ backgrounds and, for PNR, the mixing coefficient $\alpha$ . Units: $d_{i}$ and $\sigma_{i}$ are in $\mathrm{\AA}$ , all SLDs are in $10^{-6}\,\mathrm{\AA}^{-2}$ , while $r_{\mathrm{sc}}$ , the $\log_{10}$ backgrounds and $\alpha$ are dimensionless. The plot ranges correspond to the prior bounds.

Parameter type	Training range	Physical unit
Thickness	$[0,\;500]$	Å
Roughnesses	$[0,\;60]$	Å
SLDs	$[0,\;50]$	10 -6 Å -2
q-shift	$[-0.002,\;0.002]$	Å -1
Intensity scale	$[0.9,\;1.1]$	-

Table 1: Training ranges for the XRR model with standard slab parameterization

Parameter type	Training range	Physical unit
Thicknesses	$[0,\;1500]$	Å
Roughnesses	$[0,\;60]$	Å
SLDs	$[-8,\;16]$	10 -6 Å -2
Intensity scale	$[0.9,\;1.1]$	-
Background ( $\log_{10}$ )	$[-10,\;-4]$	-

Table 2: Training ranges for the NR model with standard slab parameterization

Parameter type	Training range	Physical unit
Thicknesses	$[0,\;500]$	Å
Roughnesses	$[0,\;60]$	Å
SLDs	$[-8,\;16]$	10 -6 Å -2
Solvent volume fractions	$[0,\;1]$	-
Intensity scale	$[0.9,\;1.1]$	-
Background ( $\log_{10}$ )	$[-10,\;-4]$	-

Table 3: Training ranges for the NR model with parameterization incorporating solvent penetration

Parameter type	Training range	Physical unit
Thicknesses	$[1,\;500]$	Å
Roughnesses	$[0,\;20]$	Å
Re. SLDs	$[0,\;150]$	10 -6 Å -2
Im. SLDs	$[0,\;30]$	10 -6 Å -2
Intensity scale	$[0.8,\;1.2]$	-
Background ( $\log_{10}$ )	$[-10,\;-4]$	-

Table 4: Training ranges for the XRR model with absorption parameterization

Parameter type	Training range	Physical unit
Thicknesses	$[1,\;500]$	Å
Roughnesses	$[0,\;20]$	Å
Nuclear SLDs	$[0,\;10]$	10 -6 Å -2
Magnetic SLDs	$[0,\;10]$	10 -6 Å -2
Channel mixing factor	$[0,\;0.2]$	-
Intensity scale	$[0.9,\;1.1]$	-

Table 5: Training ranges for the non–spin-flip PNR model

Parameter type	Training range	Physical unit
Thicknesses	$[1,\;500]$	Å
Nuclear SLDs	$[0,\;10]$	10 -6 Å -2
Magnetic SLDs	$[0,\;10]$	10 -6 Å -2
Magnetization angles	$[0,\;180]$	degrees

Table 6: Training ranges for the full spin-resolved PNR model

Figure 24: Relation between the real and imaginary SLD components for different elements in the periodic table and different X-ray beam energies computed using the periodictable package. The dashed line indicates the threshold used to restrict the training distribution of absorption parameters.

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University of Tennessee, Knoxville, TN, USA · 2Pennsylvania State University, University Park, PA, USA

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